:: INTEGRA6 semantic presentation begin theorem :: INTEGRA6:1 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "a")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "d"))))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set (Var "d"))) ")" )) ; theorem :: INTEGRA6:2 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) "," (Set "(" (Set (Var "x")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "b")) ")" ) "," (Set "(" (Set (Var "x")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "b")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) ; theorem :: INTEGRA6:3 (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "C")))) "holds" (Bool (Set (Set "(" (Set (Var "R")) ($#k5_relat_1 :::"|"::: ) (Set (Var "B")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "R")) ($#k5_relat_1 :::"|"::: ) (Set (Var "C")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set (Var "A")))))) ; theorem :: INTEGRA6:4 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "C")))) "holds" (Bool (Set (Set "(" ($#k7_rfunct_1 :::"chi"::: ) "(" (Set (Var "B")) "," (Set (Var "B")) ")" ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_rfunct_1 :::"chi"::: ) "(" (Set (Var "C")) "," (Set (Var "C")) ")" ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))))) ; theorem :: INTEGRA6:5 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set ($#k3_integra1 :::"vol"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a"))))) ; theorem :: INTEGRA6:6 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_integra1 :::"vol"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k3_integra5 :::"']"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" )))) ; begin theorem :: INTEGRA6:7 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set ($#k56_valued_1 :::"abs"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) "," (Set (Var "A")) ")" )) ")" ))) ; theorem :: INTEGRA6:8 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) "," (Set (Var "a")) "," (Set (Var "b")) ")" )))) ; theorem :: INTEGRA6:9 (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 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" ))) ")" )))) ; theorem :: INTEGRA6:10 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" (Set (Var "c")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ))))) ; theorem :: INTEGRA6:11 (Bool "for" (Set (Var "A")) "being" ($#~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")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "g")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k47_valued_1 :::"-"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "A")) ")" ")" ))) & (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k47_valued_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "A")) ")" ")" ))) ")" ))) ; theorem :: INTEGRA6:12 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "g")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ))) & (Bool (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k47_valued_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ))) ")" ))) ; theorem :: INTEGRA6:13 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ))) ; theorem :: INTEGRA6:14 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "g")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))))) ; theorem :: INTEGRA6:15 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k3_integra1 :::"vol"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "D"))))) "holds" (Bool (Set (Set (Var "D")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" ($#k3_integra1 :::"vol"::: ) (Set (Var "A")) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ))) ")" ) ")" )))) ; begin theorem :: INTEGRA6:16 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_integra1 :::"vol"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set ($#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 ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "Tn")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set (Var "Tn")) ($#r1_integra4 :::"divide_into_equal"::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Set (Var "T")) ($#k2_integra2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "Tn"))) ")" )) ")" ) ")" ))) ; theorem :: INTEGRA6:17 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "c")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "b")) ")" ")" ))) ")" ))) ; theorem :: INTEGRA6:18 (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) ")" ))) ; theorem :: INTEGRA6:19 (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "g")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ))) ; theorem :: INTEGRA6:20 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "c")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))))) ; theorem :: INTEGRA6:21 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k3_integra5 :::"['"::: ) (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k56_valued_1 :::"abs"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ")" )) ")" ))) ; theorem :: INTEGRA6:22 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k56_valued_1 :::"abs"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "d")) "," (Set (Var "c")) ")" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) ")" ))) ; theorem :: INTEGRA6:23 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "e"))) ")" )) "holds" (Bool "(" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "e")) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ))) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "d")) "," (Set (Var "c")) ")" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "e")) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ))) ")" ))) ; theorem :: INTEGRA6:24 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "g")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) & (Bool (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k47_valued_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) ")" ))) ; theorem :: INTEGRA6:25 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k4_integra5 :::"integral"::: ) "(" (Set "(" (Set (Var "e")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "e")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))))) ; theorem :: INTEGRA6:26 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "e")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "e"))) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "e")) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ))) ")" ))) ; theorem :: INTEGRA6:27 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "e")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "e"))) ")" ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "e")) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ))))) ; begin theorem :: INTEGRA6:28 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set (Set (Var "F")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "x")) ")" )) ")" ) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Var "F")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "F")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")))) ")" ))))) ; theorem :: INTEGRA6:29 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set (Set (Var "F")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "x")) ")" )) ")" ) & (Bool (Set (Var "F")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "F")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")))) ")" ))))) ;