:: INTEGR15 semantic presentation begin 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 ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_integra1 :::"Division"::: ) "of" (Set (Const "A")); mode :::"middle_volume"::: "of" "f" "," "D" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) means :: INTEGR15: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 "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rvsum_1 :::"rng"::: ) (Set "(" "f" ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" "D" "," (Set (Var "i")) ")" ")" ) ")" ))) & (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k3_integra1 :::"vol"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" "D" "," (Set (Var "i")) ")" ")" ) ")" ))) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"middle_volume"::: INTEGR15:def 1 : (Bool "for" (Set (Var "A")) "being" ($#~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" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k1_numbers :::"REAL"::: ) ) (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 ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m1_integr15 :::"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 "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rvsum_1 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" (Set (Var "D")) "," (Set (Var "i")) ")" ")" ) ")" ))) & (Bool (Set (Set (Var "b4")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#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 ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_integra1 :::"Division"::: ) "of" (Set (Const "A")); let "F" be ($#m1_integr15 :::"middle_volume"::: ) "of" (Set (Const "f")) "," (Set (Const "D")); func :::"middle_sum"::: "(" "f" "," "F" ")" -> ($#m1_subset_1 :::"Real":::) equals :: INTEGR15:def 2 (Set ($#k18_rvsum_1 :::"Sum"::: ) "F"); end; :: deftheorem defines :::"middle_sum"::: INTEGR15: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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "F")) "being" ($#m1_integr15 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) "holds" (Bool (Set ($#k1_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "F")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "F")))))))); theorem :: INTEGR15:1 (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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "F")) "being" ($#m1_integr15 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v2_seq_2 :::"bounded_below"::: ) )) "holds" (Bool (Set ($#k7_integra1 :::"lower_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "D")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "F")) ")" )))))) ; theorem :: INTEGR15:2 (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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "F")) "being" ($#m1_integr15 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_seq_2 :::"bounded_above"::: ) )) "holds" (Bool (Set ($#k1_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "F")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_integra1 :::"upper_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "D")) ")" )))))) ; theorem :: INTEGR15:3 (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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v2_seq_2 :::"bounded_below"::: ) ) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "e")))) "holds" (Bool "ex" (Set (Var "F")) "being" ($#m1_integr15 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) "st" (Bool (Set ($#k1_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "F")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k7_integra1 :::"lower_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "D")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set (Var "e"))))))))) ; theorem :: INTEGR15:4 (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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_seq_2 :::"bounded_above"::: ) ) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "e")))) "holds" (Bool "ex" (Set (Var "F")) "being" ($#m1_integr15 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) "st" (Bool (Set (Set "(" ($#k6_integra1 :::"upper_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "D")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "e"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (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 ($#k1_numbers :::"REAL"::: ) ); let "T" be ($#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 ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ")" ) means :: INTEGR15: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_integr15 :::"middle_volume"::: ) "of" "f" "," (Set "T" ($#k2_integra2 :::"."::: ) (Set (Var "k"))))); end; :: deftheorem defines :::"middle_volume_Sequence"::: INTEGR15:def 3 : (Bool "for" (Set (Var "A")) "being" ($#~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" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k1_numbers :::"REAL"::: ) ) (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 ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m2_integr15 :::"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_integr15 :::"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 ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); let "S" be ($#m2_integr15 :::"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" -> ($#m1_integr15 :::"middle_volume"::: ) "of" "f" "," (Set "T" ($#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 ($#k1_numbers :::"REAL"::: ) ); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); let "S" be ($#m2_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Const "f")) "," (Set (Const "T")); func :::"middle_sum"::: "(" "f" "," "S" ")" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: INTEGR15:def 4 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k1_integr15 :::"middle_sum"::: ) "(" "f" "," (Set "(" "S" ($#k2_integr15 :::"."::: ) (Set (Var "i")) ")" ) ")" ))); end; :: deftheorem defines :::"middle_sum"::: INTEGR15: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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k3_integr15 :::"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")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k1_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "S")) ($#k2_integr15 :::"."::: ) (Set (Var "i")) ")" ) ")" ))) ")" )))))); theorem :: INTEGR15:5 (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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v2_seq_2 :::"bounded_below"::: ) )) "holds" (Bool (Set (Set "(" ($#k4_integra2 :::"lower_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "T")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))))))))) ; theorem :: INTEGR15:6 (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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_seq_2 :::"bounded_above"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_integra2 :::"upper_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "T")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))))))))) ; theorem :: INTEGR15:7 (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 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "e"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v2_seq_2 :::"bounded_below"::: ) )) "holds" (Bool "ex" (Set (Var "S")) "being" ($#m2_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) "st" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k4_integra2 :::"lower_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "T")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k7_real_1 :::"+"::: ) (Set (Var "e")))))))))) ; theorem :: INTEGR15:8 (Bool "for" (Set (Var "A")) "being" ($#~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" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "e"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_seq_2 :::"bounded_above"::: ) )) "holds" (Bool "ex" (Set (Var "S")) "being" ($#m2_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) "st" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set "(" ($#k3_integra2 :::"upper_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "T")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "e"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i")))))))))) ; theorem :: INTEGR15:9 (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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_integra1 :::"integrable"::: ) ) & (Bool (Set ($#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 ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k3_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k3_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_integra1 :::"integral"::: ) (Set (Var "f")))) ")" ))))) ; theorem :: INTEGR15:10 (Bool "for" (Set (Var "A")) "being" ($#~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" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_integra1 :::"integrable"::: ) ) "iff" (Bool "ex" (Set (Var "I")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) "st" (Bool (Bool (Set ($#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 ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k3_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k3_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "I"))) ")" )))) ")" ))) ; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "D" be ($#m1_integra1 :::"Division"::: ) "of" (Set (Const "A")); mode :::"middle_volume"::: "of" "f" "," "D" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") means :: INTEGR15:def 5 (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 "r")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") "st" (Bool "(" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#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 "(" ($#k3_integra1 :::"vol"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" "D" "," (Set (Var "i")) ")" ")" ) ")" ) ($#k9_euclid :::"*"::: ) (Set (Var "r")))) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"middle_volume"::: INTEGR15:def 5 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~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" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "b5")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool "(" (Bool (Set (Var "b5")) "is" ($#m3_integr15 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b5"))) ($#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 "r")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "st" (Bool "(" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" (Set (Var "D")) "," (Set (Var "i")) ")" ")" ) ")" ))) & (Bool (Set (Set (Var "b5")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_integra1 :::"vol"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" (Set (Var "D")) "," (Set (Var "i")) ")" ")" ) ")" ) ($#k9_euclid :::"*"::: ) (Set (Var "r")))) ")" )) ")" ) ")" ) ")" )))))); definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "D" be ($#m1_integra1 :::"Division"::: ) "of" (Set (Const "A")); let "F" be ($#m3_integr15 :::"middle_volume"::: ) "of" (Set (Const "f")) "," (Set (Const "D")); func :::"middle_sum"::: "(" "f" "," "F" ")" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") means :: INTEGR15:def 6 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool "ex" (Set (Var "Fi")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set (Var "Fi")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," "n" ")" ")" ) ($#k4_finseqop :::"*"::: ) "F")) & (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "Fi")))) ")" ))); end; :: deftheorem defines :::"middle_sum"::: INTEGR15:def 6 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "F")) "being" ($#m3_integr15 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) (Bool "for" (Set (Var "b6")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool "(" (Bool (Set (Var "b6")) ($#r1_hidden :::"="::: ) (Set ($#k4_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "F")) ")" )) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool "ex" (Set (Var "Fi")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set (Var "Fi")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k4_finseqop :::"*"::: ) (Set (Var "F")))) & (Bool (Set (Set (Var "b6")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "Fi")))) ")" ))) ")" ))))))); definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); mode :::"middle_volume_Sequence"::: "of" "f" "," "T" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) means :: INTEGR15:def 7 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "k"))) "is" ($#m3_integr15 :::"middle_volume"::: ) "of" "f" "," (Set "T" ($#k2_integra2 :::"."::: ) (Set (Var "k"))))); end; :: deftheorem defines :::"middle_volume_Sequence"::: INTEGR15:def 7 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b5")) "is" ($#m4_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T"))) "iff" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b5")) ($#k3_funct_2 :::"."::: ) (Set (Var "k"))) "is" ($#m3_integr15 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Set (Var "T")) ($#k2_integra2 :::"."::: ) (Set (Var "k"))))) ")" )))))); definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); let "S" be ($#m4_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Const "f")) "," (Set (Const "T")); let "k" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); :: original: :::"."::: redefine func "S" :::"."::: "k" -> ($#m3_integr15 :::"middle_volume"::: ) "of" "f" "," (Set "T" ($#k2_integra2 :::"."::: ) "k"); end; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); let "S" be ($#m4_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Const "f")) "," (Set (Const "T")); func :::"middle_sum"::: "(" "f" "," "S" ")" -> ($#m1_subset_1 :::"sequence":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) "n" ")" ) means :: INTEGR15:def 8 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_normsp_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_integr15 :::"middle_sum"::: ) "(" "f" "," (Set "(" "S" ($#k5_integr15 :::"."::: ) (Set (Var "i")) ")" ) ")" ))); end; :: deftheorem defines :::"middle_sum"::: INTEGR15:def 8 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m4_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) (Bool "for" (Set (Var "b6")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b6")) ($#r1_hidden :::"="::: ) (Set ($#k6_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" )) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b6")) ($#k1_normsp_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "S")) ($#k5_integr15 :::"."::: ) (Set (Var "i")) ")" ) ")" ))) ")" ))))))); definitionlet "n" be ($#m1_hidden :::"Nat":::); let "Z" be ($#m1_hidden :::"set"::: ) ; let "f", "g" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); func "f" :::"+"::: "g" -> ($#m1_subset_1 :::"PartFunc":::) "of" "Z" "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) equals :: INTEGR15:def 9 (Set "f" ($#k74_valued_2 :::"<++>"::: ) "g"); func "f" :::"-"::: "g" -> ($#m1_subset_1 :::"PartFunc":::) "of" "Z" "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) equals :: INTEGR15:def 10 (Set "f" ($#k80_valued_2 :::"<-->"::: ) "g"); end; :: deftheorem defines :::"+"::: INTEGR15:def 9 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k7_integr15 :::"+"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k74_valued_2 :::"<++>"::: ) (Set (Var "g"))))))); :: deftheorem defines :::"-"::: INTEGR15:def 10 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k8_integr15 :::"-"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k80_valued_2 :::"<-->"::: ) (Set (Var "g"))))))); definitionlet "n" be ($#m1_hidden :::"Nat":::); let "r" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "Z" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); func "r" :::"(#)"::: "f" -> ($#m1_subset_1 :::"PartFunc":::) "of" "Z" "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) equals :: INTEGR15:def 11 (Set "f" ($#k43_valued_2 :::"[#]"::: ) "r"); end; :: deftheorem defines :::"(#)"::: INTEGR15:def 11 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k43_valued_2 :::"[#]"::: ) (Set (Var "r")))))))); begin definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); attr "f" is :::"bounded"::: means :: INTEGR15:def 12 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," "n" ")" ")" ) ($#k1_partfun1 :::"*"::: ) "f") "is" ($#v1_comseq_2 :::"bounded"::: ) )); end; :: deftheorem defines :::"bounded"::: INTEGR15:def 12 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~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" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_integr15 :::"bounded"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) ")" )))); definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); attr "f" is :::"integrable"::: means :: INTEGR15:def 13 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," "n" ")" ")" ) ($#k1_partfun1 :::"*"::: ) "f") "is" ($#v3_integra1 :::"integrable"::: ) )); end; :: deftheorem defines :::"integrable"::: INTEGR15:def 13 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_integr15 :::"integrable"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v3_integra1 :::"integrable"::: ) )) ")" )))); definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); func :::"integral"::: "f" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") means :: INTEGR15:def 14 (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k12_integra1 :::"integral"::: ) (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," "n" ")" ")" ) ($#k1_partfun1 :::"*"::: ) "f" ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"integral"::: INTEGR15:def 14 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k10_integr15 :::"integral"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k12_integra1 :::"integral"::: ) (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ))) ")" ) ")" ) ")" ))))); theorem :: INTEGR15:11 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~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" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m4_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_integr15 :::"bounded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_integr15 :::"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 ($#k6_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" ($#k6_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_integr15 :::"integral"::: ) (Set (Var "f")))) ")" )))))) ; theorem :: INTEGR15:12 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_integr15 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_integr15 :::"integrable"::: ) ) "iff" (Bool "ex" (Set (Var "I")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "st" (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m4_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) "st" (Bool (Bool (Set ($#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 ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k6_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" ($#k6_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "I"))) ")" )))) ")" )))) ; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); attr "f" is :::"bounded"::: means :: INTEGR15:def 15 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," "n" ")" ")" ) ($#k1_partfun1 :::"*"::: ) "f") "is" ($#v1_comseq_2 :::"bounded"::: ) )); end; :: deftheorem defines :::"bounded"::: INTEGR15:def 15 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_integr15 :::"bounded"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) ")" ))); definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "A" be ($#~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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); pred "f" :::"is_integrable_on"::: "A" means :: INTEGR15:def 16 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," "n" ")" ")" ) ($#k1_partfun1 :::"*"::: ) "f") ($#r1_integra5 :::"is_integrable_on"::: ) "A")); end; :: deftheorem defines :::"is_integrable_on"::: INTEGR15:def 16 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A")))) ")" )))); definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); func :::"integral"::: "(" "f" "," "A" ")" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") means :: INTEGR15:def 17 (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," "n" ")" ")" ) ($#k1_partfun1 :::"*"::: ) "f" ")" ) "," "A" ")" )) ")" ) ")" ); end; :: deftheorem defines :::"integral"::: INTEGR15:def 17 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" )) "iff" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) "," (Set (Var "A")) ")" )) ")" ) ")" ) ")" ))))); theorem :: INTEGR15:13 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A"))) "iff" (Bool (Set (Var "g")) "is" ($#v2_integr15 :::"integrable"::: ) ) ")" ))))) ; theorem :: INTEGR15:14 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool (Set ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_integr15 :::"integral"::: ) (Set (Var "g")))))))) ; definitionlet "a", "b" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); func :::"integral"::: "(" "f" "," "a" "," "b" ")" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") means :: INTEGR15:def 18 (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," "n" ")" ")" ) ($#k1_partfun1 :::"*"::: ) "f" ")" ) "," "a" "," "b" ")" )) ")" ) ")" ); end; :: deftheorem defines :::"integral"::: INTEGR15:def 18 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "b5")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" )) "iff" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b5"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "b5")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) "," (Set (Var "a")) "," (Set (Var "b")) ")" )) ")" ) ")" ) ")" ))))); begin theorem :: INTEGR15:15 (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f2")) ")" ))) ")" )))) ; theorem :: INTEGR15:16 (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ))))))) ; theorem :: INTEGR15:17 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "f2")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v3_integr15 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2"))) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2"))) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k11_integr15 :::"integral"::: ) "(" (Set "(" (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "f1")) "," (Set (Var "A")) ")" ")" ) ($#k7_euclid :::"+"::: ) (Set "(" ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ")" ))) & (Bool (Set ($#k11_integr15 :::"integral"::: ) "(" (Set "(" (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "f1")) "," (Set (Var "A")) ")" ")" ) ($#k8_euclid :::"-"::: ) (Set "(" ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ")" ))) ")" )))) ; theorem :: INTEGR15:18 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v3_integr15 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f"))) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k11_integr15 :::"integral"::: ) "(" (Set "(" (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k9_euclid :::"*"::: ) (Set "(" ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" ))) ")" ))))) ; theorem :: INTEGR15:19 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "A")) "being" ($#~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 ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" )))))) ; theorem :: INTEGR15:20 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (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")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k6_euclid :::"-"::: ) (Set "(" ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" )))))) ; theorem :: INTEGR15:21 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "Z")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k7_integr15 :::"+"::: ) (Set (Var "g")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_integr15 :::"+"::: ) (Set (Var "g")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k7_euclid :::"+"::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" )))))) ; theorem :: INTEGR15:22 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "Z")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k8_integr15 :::"-"::: ) (Set (Var "g")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k8_integr15 :::"-"::: ) (Set (Var "g")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" )))))) ; theorem :: INTEGR15:23 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "Z")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k9_euclid :::"*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ))))))) ;