:: INTEGR18 semantic presentation begin definitionlet "X" be ($#l1_normsp_1 :::"RealNormSpace":::); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); let "D" be ($#m1_integra1 :::"Division"::: ) "of" (Set (Const "A")); mode :::"middle_volume"::: "of" "f" "," "D" -> ($#m2_finseq_1 :::"FinSequence":::) "of" "X" means :: INTEGR18: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" ($#m1_subset_1 :::"Point":::) "of" "X" "st" (Bool "(" (Bool (Set (Var "c")) ($#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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_integra1 :::"vol"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" "D" "," (Set (Var "i")) ")" ")" ) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "c")))) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"middle_volume"::: INTEGR18:def 1 : (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~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" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (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 (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b5")) "is" ($#m1_integr18 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D"))) "iff" (Bool "(" (Bool (Set ($#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 ($#k4_finseq_1 :::"dom"::: ) (Set (Var "D"))))) "holds" (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "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 "b5")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_integra1 :::"vol"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" (Set (Var "D")) "," (Set (Var "i")) ")" ")" ) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "c")))) ")" )) ")" ) ")" ) ")" )))))); definitionlet "X" be ($#l1_normsp_1 :::"RealNormSpace":::); 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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); let "D" be ($#m1_integra1 :::"Division"::: ) "of" (Set (Const "A")); let "F" be ($#m1_integr18 :::"middle_volume"::: ) "of" (Set (Const "f")) "," (Set (Const "D")); func :::"middle_sum"::: "(" "f" "," "F" ")" -> ($#m1_subset_1 :::"Point":::) "of" "X" equals :: INTEGR18:def 2 (Set ($#k4_rlvect_1 :::"Sum"::: ) "F"); end; :: deftheorem defines :::"middle_sum"::: INTEGR18:def 2 : (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "F")) "being" ($#m1_integr18 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) "holds" (Bool (Set ($#k1_integr18 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "F")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "F"))))))))); definitionlet "X" be ($#l1_normsp_1 :::"RealNormSpace":::); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); mode :::"middle_volume_Sequence"::: "of" "f" "," "T" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ($#k3_finseq_2 :::"*"::: ) ")" ) means :: INTEGR18:def 3 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k8_nat_1 :::"."::: ) (Set (Var "k"))) "is" ($#m1_integr18 :::"middle_volume"::: ) "of" "f" "," (Set "T" ($#k2_integra2 :::"."::: ) (Set (Var "k"))))); end; :: deftheorem defines :::"middle_volume_Sequence"::: INTEGR18:def 3 : (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~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" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (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 ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ($#k3_finseq_2 :::"*"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b5")) "is" ($#m2_integr18 :::"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")) ($#k8_nat_1 :::"."::: ) (Set (Var "k"))) "is" ($#m1_integr18 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Set (Var "T")) ($#k2_integra2 :::"."::: ) (Set (Var "k"))))) ")" )))))); definitionlet "X" be ($#l1_normsp_1 :::"RealNormSpace":::); 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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); let "S" be ($#m2_integr18 :::"middle_volume_Sequence"::: ) "of" (Set (Const "f")) "," (Set (Const "T")); let "k" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); :: original: :::"."::: redefine func "S" :::"."::: "k" -> ($#m1_integr18 :::"middle_volume"::: ) "of" "f" "," (Set "T" ($#k2_integra2 :::"."::: ) "k"); end; definitionlet "X" be ($#l1_normsp_1 :::"RealNormSpace":::); 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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); let "S" be ($#m2_integr18 :::"middle_volume_Sequence"::: ) "of" (Set (Const "f")) "," (Set (Const "T")); func :::"middle_sum"::: "(" "f" "," "S" ")" -> ($#m1_subset_1 :::"sequence":::) "of" "X" means :: INTEGR18:def 4 (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 ($#k1_integr18 :::"middle_sum"::: ) "(" "f" "," (Set "(" "S" ($#k2_integr18 :::"."::: ) (Set (Var "i")) ")" ) ")" ))); end; :: deftheorem defines :::"middle_sum"::: INTEGR18:def 4 : (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~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" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr18 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) (Bool "for" (Set (Var "b6")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b6")) ($#r1_hidden :::"="::: ) (Set ($#k3_integr18 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" )) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b6")) ($#k1_normsp_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k1_integr18 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "S")) ($#k2_integr18 :::"."::: ) (Set (Var "i")) ")" ) ")" ))) ")" ))))))); begin definitionlet "X" be ($#l1_normsp_1 :::"RealNormSpace":::); 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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); attr "f" is :::"integrable"::: means :: INTEGR18:def 5 (Bool "ex" (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" "X" "st" (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" "A" (Bool "for" (Set (Var "S")) "being" ($#m2_integr18 :::"middle_volume_Sequence"::: ) "of" "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 ($#k3_integr18 :::"middle_sum"::: ) "(" "f" "," (Set (Var "S")) ")" ) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" ($#k3_integr18 :::"middle_sum"::: ) "(" "f" "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "I"))) ")" )))); end; :: deftheorem defines :::"integrable"::: INTEGR18:def 5 : (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_integr18 :::"integrable"::: ) ) "iff" (Bool "ex" (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr18 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) "st" (Bool (Bool (Set ($#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_integr18 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" ($#k3_integr18 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "I"))) ")" )))) ")" )))); theorem :: INTEGR18:1 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "R1")) "," (Set (Var "R2")) "," (Set (Var "R3")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "R1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "R2")))) & (Bool (Set (Var "R3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "R1")) ($#k1_binom :::"+"::: ) (Set (Var "R2"))))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "R3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "R1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "R2")) ")" ))))) ; theorem :: INTEGR18:2 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "R1")) "," (Set (Var "R2")) "," (Set (Var "R3")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "R1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "R2")))) & (Bool (Set (Var "R3")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "R1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "R2"))))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "R3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "R1")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "R2")) ")" ))))) ; theorem :: INTEGR18:3 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "R1")) "," (Set (Var "R2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "R2")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "a")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "R1"))))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "R2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "R1")) ")" )))))) ; definitionlet "X" be ($#l1_normsp_1 :::"RealNormSpace":::); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); assume (Bool (Set (Const "f")) "is" ($#v1_integr18 :::"integrable"::: ) ) ; func :::"integral"::: "f" -> ($#m1_subset_1 :::"Point":::) "of" "X" means :: INTEGR18:def 6 (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" "A" (Bool "for" (Set (Var "S")) "being" ($#m2_integr18 :::"middle_volume_Sequence"::: ) "of" "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 ($#k3_integr18 :::"middle_sum"::: ) "(" "f" "," (Set (Var "S")) ")" ) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" ($#k3_integr18 :::"middle_sum"::: ) "(" "f" "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) it) ")" ))); end; :: deftheorem defines :::"integral"::: INTEGR18:def 6 : (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_integr18 :::"integrable"::: ) )) "holds" (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k4_integr18 :::"integral"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr18 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) "st" (Bool (Bool (Set ($#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_integr18 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" ($#k3_integr18 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "b4"))) ")" ))) ")" ))))); theorem :: INTEGR18:4 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~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 "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Var "h")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v1_integr18 :::"integrable"::: ) )) "holds" (Bool "(" (Bool (Set (Var "h")) "is" ($#v1_integr18 :::"integrable"::: ) ) & (Bool (Set ($#k4_integr18 :::"integral"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set "(" ($#k4_integr18 :::"integral"::: ) (Set (Var "f")) ")" ))) ")" ))))) ; theorem :: INTEGR18:5 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Var "h")) ($#r2_relset_1 :::"="::: ) (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v1_integr18 :::"integrable"::: ) )) "holds" (Bool "(" (Bool (Set (Var "h")) "is" ($#v1_integr18 :::"integrable"::: ) ) & (Bool (Set ($#k4_integr18 :::"integral"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k4_integr18 :::"integral"::: ) (Set (Var "f")) ")" ))) ")" )))) ; theorem :: INTEGR18:6 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Var "h")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) "is" ($#v1_integr18 :::"integrable"::: ) ) & (Bool (Set (Var "g")) "is" ($#v1_integr18 :::"integrable"::: ) )) "holds" (Bool "(" (Bool (Set (Var "h")) "is" ($#v1_integr18 :::"integrable"::: ) ) & (Bool (Set ($#k4_integr18 :::"integral"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_integr18 :::"integral"::: ) (Set (Var "f")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k4_integr18 :::"integral"::: ) (Set (Var "g")) ")" ))) ")" )))) ; theorem :: INTEGR18:7 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Var "h")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) "is" ($#v1_integr18 :::"integrable"::: ) ) & (Bool (Set (Var "g")) "is" ($#v1_integr18 :::"integrable"::: ) )) "holds" (Bool "(" (Bool (Set (Var "h")) "is" ($#v1_integr18 :::"integrable"::: ) ) & (Bool (Set ($#k4_integr18 :::"integral"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_integr18 :::"integral"::: ) (Set (Var "f")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k4_integr18 :::"integral"::: ) (Set (Var "g")) ")" ))) ")" )))) ; definitionlet "X" be ($#l1_normsp_1 :::"RealNormSpace":::); 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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); pred "f" :::"is_integrable_on"::: "A" means :: INTEGR18:def 7 (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" "A" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set "f" ($#k2_partfun1 :::"|"::: ) "A")) & (Bool (Set (Var "g")) "is" ($#v1_integr18 :::"integrable"::: ) ) ")" )); end; :: deftheorem defines :::"is_integrable_on"::: INTEGR18:def 7 : (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A")))) & (Bool (Set (Var "g")) "is" ($#v1_integr18 :::"integrable"::: ) ) ")" )) ")" )))); definitionlet "X" be ($#l1_normsp_1 :::"RealNormSpace":::); let "A" be ($#~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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); assume (Bool (Set (Const "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Const "f")))) ; func :::"integral"::: "(" "f" "," "A" ")" -> ($#m1_subset_1 :::"Element":::) "of" "X" means :: INTEGR18:def 8 (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" "A" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set "f" ($#k2_partfun1 :::"|"::: ) "A")) & (Bool it ($#r1_hidden :::"="::: ) (Set ($#k4_integr18 :::"integral"::: ) (Set (Var "g")))) ")" )); end; :: deftheorem defines :::"integral"::: INTEGR18:def 8 : (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" )) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A")))) & (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k4_integr18 :::"integral"::: ) (Set (Var "g")))) ")" )) ")" ))))); theorem :: INTEGR18:8 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) "iff" (Bool (Set (Var "g")) "is" ($#v1_integr18 :::"integrable"::: ) ) ")" ))))) ; theorem :: INTEGR18:9 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool (Set ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_integr18 :::"integral"::: ) (Set (Var "g")))))))) ; theorem :: INTEGR18:10 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "g1")) "," (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Y")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "g1"))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "f1")))) "holds" (Bool (Set (Set (Var "g1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f")))))))) ; theorem :: INTEGR18:11 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "g1")) "," (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Y")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "g1"))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "f1")))) "holds" (Bool (Set (Set (Var "g1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f")))))))) ; theorem :: INTEGR18:12 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Y")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "g1")))) "holds" (Bool (Set (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "g1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "g"))))))))) ; begin theorem :: INTEGR18:13 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "r")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k5_integr18 :::"integral"::: ) "(" (Set "(" (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set "(" ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" ))) ")" ))))) ; theorem :: INTEGR18:14 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~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 "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Var "f1")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "f2")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k5_integr18 :::"integral"::: ) "(" (Set "(" (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f1")) "," (Set (Var "A")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ")" ))) ")" )))) ; theorem :: INTEGR18:15 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "st" (Bool (Bool (Set (Var "f1")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "f2")) ($#r1_integr18 :::"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"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k5_integr18 :::"integral"::: ) "(" (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f1")) "," (Set (Var "A")) ")" ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ")" ))) ")" )))) ; definitionlet "X" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); let "a", "b" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"integral"::: "(" "f" "," "a" "," "b" ")" -> ($#m1_subset_1 :::"Element":::) "of" "X" equals :: INTEGR18:def 9 (Set ($#k5_integr18 :::"integral"::: ) "(" "f" "," (Set ($#k3_integra5 :::"['"::: ) "a" "," "b" ($#k3_integra5 :::"']"::: ) ) ")" ) if (Bool "a" ($#r1_xxreal_0 :::"<="::: ) "b") otherwise (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k5_integr18 :::"integral"::: ) "(" "f" "," (Set ($#k3_integra5 :::"['"::: ) "b" "," "a" ($#k3_integra5 :::"']"::: ) ) ")" ")" )); end; :: deftheorem defines :::"integral"::: INTEGR18:def 9 : (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "implies" (Bool (Set ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ")" )) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))))) "implies" (Bool (Set ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set ($#k3_integra5 :::"['"::: ) (Set (Var "b")) "," (Set (Var "a")) ($#k3_integra5 :::"']"::: ) ) ")" ")" ))) ")" ")" )))); theorem :: INTEGR18:16 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (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")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" )))))) ; theorem :: INTEGR18:17 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_integra1 :::"vol"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "X")))) ")" )))) ; theorem :: INTEGR18:18 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (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")) ($#k1_rcomp_1 :::".]"::: ) )) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" )))))) ;