:: INTEGR12 semantic presentation begin theorem :: INTEGR12:1 (Bool "for" (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 "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ))) & (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 "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool (Set (Var "f2")) ($#r2_relset_1 :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2)))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k7_binop_2 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set "(" (Num 1) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k2_newton :::"|^"::: ) (Num 2) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" ))) ; theorem :: INTEGR12:2 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g1")) "," (Set (Var "g2")) "," (Set (Var "f2")) "being" ($#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")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "g1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g2")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")))) & (Bool (Set (Var "f2")) ($#r2_relset_1 :::"="::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "g2")) ($#r2_relset_1 :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (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 "g1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:3 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#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 "(" "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 ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:4 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:5 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:6 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:7 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:8 (Bool "for" (Set (Var "A")) "being" ($#~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 "f1")) "," (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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:9 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:10 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"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 "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:11 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "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 ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:12 (Bool "for" (Set (Var "A")) "being" ($#~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 "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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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 ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:13 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:14 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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 "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:15 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g")) "," (Set (Var "f")) "being" ($#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 (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ))) & (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k12_binop_2 :::"/"::: ) (Set (Var "r")))) & (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k19_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "f2")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "r")) ($#k12_binop_2 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "r")) ($#k12_binop_2 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:16 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g")) "," (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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ))) & (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k12_binop_2 :::"/"::: ) (Set (Var "r")))) & (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k19_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "f2")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "r")) ($#k12_binop_2 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "r")) ($#k12_binop_2 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:17 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "g")) "being" ($#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")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "g")) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "f1")) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k12_binop_2 :::"/"::: ) (Set (Var "r")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k19_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:18 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "g")) "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 "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "g")) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "f1")) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) & (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 "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k12_binop_2 :::"/"::: ) (Set (Var "r")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k19_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" ) & (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 "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:19 (Bool "for" (Set (Var "n")) "being" ($#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"::: ) ) (Bool "for" (Set (Var "f1")) "," (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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:20 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#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 (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (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 "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:21 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:22 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#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 "(" "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 "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:23 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f")) "being" ($#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 (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (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 "(" (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set (Var "b")))) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:24 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f")) "being" ($#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 (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (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 "(" (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set (Var "b")))) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:25 (Bool "for" (Set (Var "a")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "g")) "," (Set (Var "f1")) "," (Set (Var "f")) "being" ($#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 "(" "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 "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k12_binop_2 :::"/"::: ) (Set (Var "a")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k19_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" ) & (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"::: ) ) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "g")) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f1")) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:26 (Bool "for" (Set (Var "a")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "g")) "," (Set (Var "f1")) "," (Set (Var "f")) "being" ($#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 "(" "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 "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k12_binop_2 :::"/"::: ) (Set (Var "a")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k19_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" ) & (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"::: ) ) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "g")) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f1")) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:27 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set (Var "n")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:28 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set (Var "n")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:29 (Bool "for" (Set (Var "a")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#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 (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k41_valued_1 :::"^2"::: ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k41_valued_1 :::"^2"::: ) ")" ))) & (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")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")))) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k12_binop_2 :::"/"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k12_binop_2 :::"/"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:30 (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 "f1")) "," (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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k41_valued_1 :::"^2"::: ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k41_valued_1 :::"^2"::: ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")))) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k7_binop_2 :::"-"::: ) (Set "(" (Num 1) ($#k12_binop_2 :::"/"::: ) (Set (Var "a")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k7_binop_2 :::"-"::: ) (Set "(" (Num 1) ($#k12_binop_2 :::"/"::: ) (Set (Var "a")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:31 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#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 "(" "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 "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set (Var "b")))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:32 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "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 "f1")) "," (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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set (Var "b")))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:33 (Bool "for" (Set (Var "A")) "being" ($#~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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:34 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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 "(" "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 "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:35 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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 "(" "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 (Num 1) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k11_binop_2 :::"*"::: ) (Set "(" (Num 1) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) & (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k19_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ))) & (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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:36 (Bool "for" (Set (Var "A")) "being" ($#~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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"))) ($#r1_hidden :::"="::: ) (Set ($#k7_binop_2 :::"-"::: ) (Set "(" (Num 1) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k11_binop_2 :::"*"::: ) (Set "(" (Num 1) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) & (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k19_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ))) & (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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:37 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "being" ($#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")) ($#k12_binop_2 :::"/"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set (Var "b")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set (Var "b")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k19_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "Z")) ($#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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:38 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "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")) "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"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k12_binop_2 :::"/"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set (Var "b")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set (Var "b")))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k19_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (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 "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:39 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "g")) "," (Set (Var "f2")) "," (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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f1")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "g")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")))) & (Bool (Set (Var "f2")) ($#r2_relset_1 :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k5_prepower :::"#Z"::: ) (Num 2) ")" ) ")" ) ($#k9_prepower :::"#R"::: ) (Set "(" ($#k13_binop_2 :::"-"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ")" ))) & (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "Z")) ($#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 "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:40 (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 "g")) "," (Set (Var "f1")) "," (Set (Var "f2")) "," (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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "g")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")))) & (Bool (Set (Var "f2")) ($#r2_relset_1 :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "a")) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k5_prepower :::"#Z"::: ) (Num 2) ")" ) ")" ) ($#k9_prepower :::"#R"::: ) (Set "(" ($#k13_binop_2 :::"-"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ")" ))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k5_square_1 :::"^2"::: ) )) & (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ))) & (Bool (Set (Var "Z")) ($#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 "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:41 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"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 "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (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")) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ))) & (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k13_binop_2 :::"-"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Set (Var "n")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k13_binop_2 :::"-"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Set (Var "n")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:42 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"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 "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (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")) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ))) & (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR12:43 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g1")) "," (Set (Var "g2")) "," (Set (Var "f2")) "being" ($#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")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "g1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g2")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")))) & (Bool (Set (Var "f2")) ($#r2_relset_1 :::"="::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "g2")) ($#r2_relset_1 :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (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 (Num 1) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set "(" (Num 1) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) & (Bool (Set (Set (Var "g1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:44 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#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 (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (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 ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k6_rfunct_1 :::"^"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR12:45 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k19_binop_2 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ($#k6_rfunct_1 :::"^"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ;