:: INTEGR19 semantic presentation begin theorem :: INTEGR19:1 (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) ")" )) ; theorem :: INTEGR19:2 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "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")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "X"))))) ; theorem :: INTEGR19:3 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (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")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#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")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (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 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "X"))))) ; theorem :: INTEGR19:4 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "b")) "being" ($#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_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "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")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k7_integr15 :::"+"::: ) (Set (Var "g")) ")" )))))) ; theorem :: INTEGR19:5 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#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 ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k8_integr15 :::"-"::: ) (Set (Var "g")) ")" )))))) ; theorem :: INTEGR19:6 (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#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 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ))) ; theorem :: INTEGR19:7 (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "b")) "being" ($#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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k47_valued_1 :::"-"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k47_valued_1 :::"-"::: ) (Set (Var "g")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ))) ; theorem :: INTEGR19:8 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "c")) ")" ")" ) ($#k7_euclid :::"+"::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "b")) ")" ")" ))) ")" )))) ; theorem :: INTEGR19:9 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "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_integr15 :::"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")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) ")" )))) ; theorem :: INTEGR19:10 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "b")) "being" ($#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_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "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_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "g")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_integr15 :::"+"::: ) (Set (Var "g"))) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_integr15 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) ")" )))) ; theorem :: INTEGR19:11 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#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")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f"))) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set "(" (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) ")" )))) ; theorem :: INTEGR19:12 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#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")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f"))) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set "(" ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) ")" )))) ; theorem :: INTEGR19:13 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#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")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "g")) ($#r1_integr15 :::"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")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_integr15 :::"-"::: ) (Set (Var "g"))) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k8_integr15 :::"-"::: ) (Set (Var "g")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) ")" )))) ; theorem :: INTEGR19:14 (Bool "for" (Set (Var "A")) "being" ($#~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 "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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" ($#v1_integr15 :::"bounded"::: ) ) "iff" (Bool (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" )))) ; theorem :: INTEGR19:15 (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")) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v3_integr15 :::"bounded"::: ) )))) ; theorem :: INTEGR19:16 (Bool "for" (Set (Var "n")) "being" ($#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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool (Set (Var "g")) "is" ($#v1_integr15 :::"bounded"::: ) ))))) ; theorem :: INTEGR19:17 (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 "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "g")) ($#k1_nfcont_4 :::".|"::: ) )))))) ; theorem :: INTEGR19:18 (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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#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 "h"))))) "holds" (Bool (Set ($#k1_nfcont_4 :::"|."::: ) (Set "(" (Set (Var "h")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A")) ")" ) ($#k1_nfcont_4 :::".|"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "h")) ($#k1_nfcont_4 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))))))) ; theorem :: INTEGR19:19 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "h")))) & (Bool (Set (Set (Var "h")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v3_integr15 :::"bounded"::: ) )) "holds" (Bool (Set (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "h")) ($#k1_nfcont_4 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )))) ; theorem :: INTEGR19:20 (Bool "for" (Set (Var "A")) "being" ($#~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" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "h")))) & (Bool (Set (Set (Var "h")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set (Var "h")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "h")) ($#k1_nfcont_4 :::".|"::: ) ) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "h")) "," (Set (Var "A")) ")" ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_integra5 :::"integral"::: ) "(" (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "h")) ($#k1_nfcont_4 :::".|"::: ) ) "," (Set (Var "A")) ")" ))))) ; theorem :: INTEGR19:21 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h")))) & (Bool (Set (Var "h")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "h")) ($#k1_nfcont_4 :::".|"::: ) ) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set (Var "h")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) )) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "h")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "h")) ($#k1_nfcont_4 :::".|"::: ) ) "," (Set (Var "a")) "," (Set (Var "b")) ")" ))))) ; theorem :: INTEGR19:22 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#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")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#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")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#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 ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set ($#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 ($#k12_euclid :::"|."::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) "," (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ")" )) ")" )))) ; theorem :: INTEGR19:23 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#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")) ")" ) "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")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#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")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"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 ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#r1_integra5 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "d")) "," (Set (Var "c")) ")" ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_integra5 :::"integral"::: ) "(" (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) ")" )))) ; theorem :: INTEGR19:24 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#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")) ")" ) "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")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#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")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"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")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "e"))) ")" )) "holds" (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "e")) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "d")) ($#k10_binop_2 :::"-"::: ) (Set (Var "c")) ")" ))) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "d")) "," (Set (Var "c")) ")" ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "e")) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "d")) ($#k10_binop_2 :::"-"::: ) (Set (Var "c")) ")" ))) ")" )))) ; theorem :: INTEGR19:25 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"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")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"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 ($#k12_integr15 :::"integral"::: ) "(" (Set "(" (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k9_euclid :::"*"::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )))))) ; theorem :: INTEGR19:26 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"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")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"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 ($#k12_integr15 :::"integral"::: ) "(" (Set "(" ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_euclid :::"-"::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )))))) ; theorem :: INTEGR19:27 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#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_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "g")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "c")) ($#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")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k12_integr15 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_integr15 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k7_euclid :::"+"::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )))))) ; theorem :: INTEGR19:28 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#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_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "g")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "c")) ($#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")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k12_integr15 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k8_integr15 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k8_euclid :::"-"::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )))))) ; theorem :: INTEGR19:29 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "E")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "E"))) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "b")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" ) ($#k9_euclid :::"*"::: ) (Set (Var "E")))) ")" ))))) ; theorem :: INTEGR19:30 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "E")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "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")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "E"))) ")" ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"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 ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "d")) ($#k10_binop_2 :::"-"::: ) (Set (Var "c")) ")" ) ($#k9_euclid :::"*"::: ) (Set (Var "E")))))))) ; theorem :: INTEGR19:31 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#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")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "c")) ")" ")" ) ($#k7_euclid :::"+"::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )))))) ; theorem :: INTEGR19:32 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#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" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#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 ($#k3_integra5 :::"['"::: ) (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "e"))) ")" )) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "n")) ($#k11_binop_2 :::"*"::: ) (Set (Var "e")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "d")) ($#k10_binop_2 :::"-"::: ) (Set (Var "c")) ")" ) ")" )))))) ; theorem :: INTEGR19:33 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b")) "," (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "b")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_euclid :::"-"::: ) (Set "(" ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" )))))) ; begin definitionlet "R" be ($#l1_normsp_1 :::"RealNormSpace":::); let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "g" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "X")) "," (Set (Const "R")); attr "g" is :::"bounded"::: means :: INTEGR19:def 1 (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "g"))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" "g" ($#k7_partfun1 :::"/."::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))))); end; :: deftheorem defines :::"bounded"::: INTEGR19:def 1 : (Bool "for" (Set (Var "R")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "X")) "," (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v1_integr19 :::"bounded"::: ) ) "iff" (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))))) ")" )))); theorem :: INTEGR19:34 (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_integr15 :::"bounded"::: ) ) "iff" (Bool (Set (Var "g")) "is" ($#v1_integr19 :::"bounded"::: ) ) ")" )))) ; theorem :: INTEGR19:35 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_integr19 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v1_integr19 :::"bounded"::: ) ) ")" )))) ; theorem :: INTEGR19:36 (Bool "for" (Set (Var "n")) "being" ($#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 "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) (Bool "for" (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#m3_integr15 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D"))) "iff" (Bool (Set (Var "q")) "is" ($#m1_integr18 :::"middle_volume"::: ) "of" (Set (Var "g")) "," (Set (Var "D"))) ")" )))))))) ; theorem :: INTEGR19:37 (Bool "for" (Set (Var "n")) "being" ($#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 "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "p")) "being" ($#m3_integr15 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) (Bool "for" (Set (Var "q")) "being" ($#m1_integr18 :::"middle_volume"::: ) "of" (Set (Var "g")) "," (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) ($#r1_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k4_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_integr18 :::"middle_sum"::: ) "(" (Set (Var "g")) "," (Set (Var "q")) ")" ))))))))) ; theorem :: INTEGR19:38 (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" )) ($#k3_finseq_2 :::"*"::: ) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "p")) ($#r1_funct_2 :::"="::: ) (Set (Var "q")))) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#m4_integr15 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T"))) "iff" (Bool (Set (Var "q")) "is" ($#m2_integr18 :::"middle_volume_Sequence"::: ) "of" (Set (Var "g")) "," (Set (Var "T"))) ")" )))))))) ; theorem :: INTEGR19:39 (Bool "for" (Set (Var "n")) "being" ($#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 "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (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 "U")) "being" ($#m2_integr18 :::"middle_volume_Sequence"::: ) "of" (Set (Var "g")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r1_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "S")) ($#r1_funct_2 :::"="::: ) (Set (Var "U")))) "holds" (Bool (Set ($#k6_integr15 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) ($#r2_funct_2 :::"="::: ) (Set ($#k3_integr18 :::"middle_sum"::: ) "(" (Set (Var "g")) "," (Set (Var "U")) ")" ))))))))) ; theorem :: INTEGR19:40 (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "I")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) (Bool "for" (Set (Var "J")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Var "J")))) "holds" (Bool "(" (Bool "(" "for" (Set (Var "T")) "being" ($#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"))) ")" )) ")" ) "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 "g")) "," (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 ($#k3_integr18 :::"middle_sum"::: ) "(" (Set (Var "g")) "," (Set (Var "S")) ")" ) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" ($#k3_integr18 :::"middle_sum"::: ) "(" (Set (Var "g")) "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "J"))) ")" ))) ")" ))))))) ; theorem :: INTEGR19:41 (Bool "for" (Set (Var "n")) "being" ($#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 "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "f")) "is" ($#v1_integr15 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_integr15 :::"integrable"::: ) ) "iff" (Bool (Set (Var "g")) "is" ($#v1_integr18 :::"integrable"::: ) ) ")" ))))) ; theorem :: INTEGR19:42 (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 "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "f")) "is" ($#v1_integr15 :::"bounded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_integr15 :::"integrable"::: ) )) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v1_integr18 :::"integrable"::: ) ) & (Bool (Set ($#k10_integr15 :::"integral"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_integr18 :::"integral"::: ) (Set (Var "g")))) ")" ))))) ; theorem :: INTEGR19:43 (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" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A"))) "iff" (Bool (Set (Var "g")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) ")" ))))) ; theorem :: INTEGR19:44 (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" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set (Var "A")))) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k11_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_integr18 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "A")) ")" )) ")" ))))) ; theorem :: INTEGR19:45 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v3_integr15 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_integr15 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "a")) "," (Set (Var "b")) ")" )))))) ; theorem :: INTEGR19:46 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "g")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set ($#k6_integr18 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ))) & (Bool (Set ($#k6_integr18 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ))) ")" )))) ; theorem :: INTEGR19:47 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "b")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" )))))) ; theorem :: INTEGR19:48 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (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 ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_integr15 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))))) ; theorem :: INTEGR19:49 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "g")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#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" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "c")) ($#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")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k6_integr18 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )))))) ; theorem :: INTEGR19:50 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) & (Bool (Set (Var "g")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#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" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) )) "is" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "c")) ($#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")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k6_integr18 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "g")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )))))) ; theorem :: INTEGR19:51 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "E")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "E"))) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#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" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "b")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "E")))) ")" ))))) ; theorem :: INTEGR19:52 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "E")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "E"))) ")" ) & (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 ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "d")) ($#k10_binop_2 :::"-"::: ) (Set (Var "c")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "E")))))))) ; theorem :: INTEGR19:53 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#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" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#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")) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "c")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )))))) ; theorem :: INTEGR19:54 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#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" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "c")) ($#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 ($#k3_integra5 :::"['"::: ) (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k3_integra5 :::"']"::: ) ))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "e"))) ")" )) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "n")) ($#k11_binop_2 :::"*"::: ) (Set (Var "e")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "d")) ($#k10_binop_2 :::"-"::: ) (Set (Var "c")) ")" ) ")" )))))) ; begin theorem :: INTEGR19:55 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "F")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr18 :::"is_integrable_on"::: ) (Set ($#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" ($#v1_integr19 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "x")) ")" )) ")" ) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Var "F")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "F")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")))) ")" )))) ; theorem :: INTEGR19:56 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "f")) ($#r1_integr18 :::"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_integr19 :::"bounded"::: ) ) & (Bool (Set ($#k3_integra5 :::"['"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_integra5 :::"']"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"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_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "F")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_integr18 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "x")) ")" )) ")" ) & (Bool (Set (Var "F")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "F")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")))) ")" ))))) ;