:: INTEGRA9 semantic presentation begin theorem :: INTEGRA9:1 (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k26_sin_cos :::"exp_R"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set (Var "x")) ")" ))) ")" ) ")" ) ; theorem :: INTEGRA9:2 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "r")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "r")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "r")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "r")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "r")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" )) ; theorem :: INTEGRA9:3 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "r")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "r")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "r")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "r")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "r")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGRA9:4 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k18_sin_cos :::"sin"::: ) (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: INTEGRA9:5 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGRA9:6 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k21_sin_cos :::"cos"::: ) (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: INTEGRA9:7 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGRA9:8 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z")))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGRA9:9 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z")))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGRA9:10 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGRA9:11 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGRA9:12 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z")))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGRA9:13 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGRA9:14 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z")))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGRA9:15 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z")))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGRA9:16 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z")))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGRA9:17 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))) ; theorem :: INTEGRA9:18 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k5_prepower :::"#Z"::: ) (Set (Var "n")))) ")" ) ")" )) ; theorem :: INTEGRA9:19 (Bool "for" (Set (Var "n")) "being" ($#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"::: ) ) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ($#k2_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ($#k2_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))))) ; begin theorem :: INTEGRA9:20 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k47_valued_1 :::"-"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "C"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_integra5 :::"||"::: ) (Set (Var "C")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "g")) ($#k1_integra5 :::"||"::: ) (Set (Var "C")) ")" ))))) ; theorem :: INTEGRA9:21 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "C")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "g")) ($#k1_integra5 :::"||"::: ) (Set (Var "C")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "C")))))) ; theorem :: INTEGRA9:22 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "C")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "g")) ($#k1_integra5 :::"||"::: ) (Set (Var "C")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "C")))))) ; theorem :: INTEGRA9:23 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "C")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "g")) ($#k1_integra5 :::"||"::: ) (Set (Var "C")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k1_integra5 :::"||"::: ) (Set (Var "C")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f2")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "C")) ")" ))))) ; definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f", "g" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); func :::"|||(":::"f" "," "g" "," "A":::")|||"::: -> ($#m1_subset_1 :::"Real":::) equals :: INTEGRA9:def 1 (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" "f" ($#k20_valued_1 :::"(#)"::: ) "g" ")" ) "," "A" ")" ); end; :: deftheorem defines :::"|||("::: INTEGRA9:def 1 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) "," (Set (Var "A")) ")" )))); theorem :: INTEGRA9:24 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) )))) ; theorem :: INTEGRA9:25 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set (Var "f2")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v3_integra1 :::"integrable"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f2")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set (Var "f2")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v3_integra1 :::"integrable"::: ) )) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f1")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#k7_real_1 :::"+"::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f2")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ))))) ; theorem :: INTEGRA9:26 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set (Var "f2")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v3_integra1 :::"integrable"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f2")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set (Var "f2")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v3_integra1 :::"integrable"::: ) )) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f1")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#k9_real_1 :::"-"::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f2")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ))))) ; theorem :: INTEGRA9:27 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" )))) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ))))) ; theorem :: INTEGRA9:28 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" )))) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) )))))) ; theorem :: INTEGRA9:29 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" )))) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "p")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set (Var "p")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) )))))) ; theorem :: INTEGRA9:30 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) "," (Set (Var "h")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "h")) ")" ) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) )))) ; theorem :: INTEGRA9:31 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set "(" (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set "(" (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "f")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "g")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ))))) ; begin definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f", "g" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); pred "f" :::"is_orthogonal_with"::: "g" "," "A" means :: INTEGRA9:def 2 (Bool (Set ($#k1_integra9 :::"|||("::: ) "f" "," "g" "," "A" ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )); end; :: deftheorem defines :::"is_orthogonal_with"::: INTEGRA9:def 2 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set (Var "g")) "," (Set (Var "A"))) "iff" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))); theorem :: INTEGRA9:32 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "f")) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set (Var "g")) "," (Set (Var "A")))) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set "(" (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set "(" (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "f")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#k7_real_1 :::"+"::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "g")) "," (Set (Var "g")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ))))) ; theorem :: INTEGRA9:33 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v3_integra1 :::"integrable"::: ) ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "f")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: INTEGRA9:34 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k16_sin_cos :::"sin"::: ) ) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "A")))) ; theorem :: INTEGRA9:35 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k8_real_1 :::"*"::: ) (Num 2) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k16_sin_cos :::"sin"::: ) ) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "A")))) ; theorem :: INTEGRA9:36 (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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k16_sin_cos :::"sin"::: ) ) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "A"))))) ; theorem :: INTEGRA9:37 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#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"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ")" ) "," (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k16_sin_cos :::"sin"::: ) ) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "A")))))) ; theorem :: INTEGRA9:38 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k16_sin_cos :::"sin"::: ) ) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "A")))) ; theorem :: INTEGRA9:39 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k16_sin_cos :::"sin"::: ) ) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "A")))) ; theorem :: INTEGRA9:40 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k16_sin_cos :::"sin"::: ) ) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "A")))) ; theorem :: INTEGRA9:41 (Bool "for" (Set (Var "n")) "being" ($#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"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k16_sin_cos :::"sin"::: ) ) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "A"))))) ; theorem :: INTEGRA9:42 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ")" ) "," (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k16_sin_cos :::"sin"::: ) ) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "A")))))) ; begin definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); func :::"||..":::"f" "," "A":::"..||"::: -> ($#m1_subset_1 :::"Real":::) equals :: INTEGRA9:def 3 (Set ($#k7_square_1 :::"sqrt"::: ) (Set ($#k1_integra9 :::"|||("::: ) "f" "," "f" "," "A" ($#k1_integra9 :::")|||"::: ) )); end; :: deftheorem defines :::"||.."::: INTEGRA9:def 3 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k2_integra9 :::"||.."::: ) (Set (Var "f")) "," (Set (Var "A")) ($#k2_integra9 :::"..||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_square_1 :::"sqrt"::: ) (Set ($#k1_integra9 :::"|||("::: ) (Set (Var "f")) "," (Set (Var "f")) "," (Set (Var "A")) ($#k1_integra9 :::")|||"::: ) ))))); theorem :: INTEGRA9:43 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_integra9 :::"||.."::: ) (Set (Var "f")) "," (Set (Var "A")) ($#k2_integra9 :::"..||"::: ) )))) ; theorem :: INTEGRA9:44 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k2_integra9 :::"||.."::: ) (Set "(" (Num 1) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "A")) ($#k2_integra9 :::"..||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_integra9 :::"||.."::: ) (Set (Var "f")) "," (Set (Var "A")) ($#k2_integra9 :::"..||"::: ) )))) ; theorem :: INTEGRA9:45 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A"))) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "f")) ($#r1_integra9 :::"is_orthogonal_with"::: ) (Set (Var "g")) "," (Set (Var "A"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_integra5 :::"||"::: ) (Set (Var "A")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set ($#k2_integra9 :::"||.."::: ) (Set "(" (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g")) ")" ) "," (Set (Var "A")) ($#k2_integra9 :::"..||"::: ) ) ($#k5_square_1 :::"^2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k2_integra9 :::"||.."::: ) (Set (Var "f")) "," (Set (Var "A")) ($#k2_integra9 :::"..||"::: ) ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k2_integra9 :::"||.."::: ) (Set (Var "g")) "," (Set (Var "A")) ($#k2_integra9 :::"..||"::: ) ) ($#k5_square_1 :::"^2"::: ) ")" ))))) ; begin theorem :: INTEGRA9:46 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k1_real_1 :::"-"::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set (Var "a")) ")" ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; theorem :: INTEGRA9:47 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:48 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:49 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:50 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:51 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:52 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:53 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:54 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:55 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "a")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "a")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:56 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:57 (Bool "for" (Set (Var "b")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "b")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "b")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "b")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "b")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "b")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:58 (Bool "for" (Set (Var "b")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "b")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "b")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "b")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "b")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "b")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:59 (Bool "for" (Set (Var "b")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "b")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "b")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "b")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "b")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "b")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:60 (Bool "for" (Set (Var "b")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "b")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "b")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:61 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ))) & (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGRA9:62 (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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "x")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k10_real_1 :::"/"::: ) (Set (Var "x")))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:63 (Bool "for" (Set (Var "A")) "being" ($#~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 "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGRA9:64 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")) ")" ) ($#k10_prepower :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:65 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")) ")" ) ($#k10_prepower :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:66 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")) ")" ) ($#k10_prepower :::"#R"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:67 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")) ")" ) ($#k10_prepower :::"#R"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ))) ")" ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGRA9:68 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGRA9:69 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGRA9:70 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: INTEGRA9:71 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ;