:: INTEGR14 semantic presentation begin theorem :: INTEGR14:1 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:2 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:3 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:4 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:5 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" )) ; theorem :: INTEGR14:6 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "Z")) "being" ($#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 "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k4_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" ) ")" ))) ; theorem :: INTEGR14:7 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:8 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:9 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:10 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:11 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:12 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:13 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR14:14 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; begin theorem :: INTEGR14:15 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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 ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#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 "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:16 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:17 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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 ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ))) & (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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:18 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:19 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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 ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:20 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:21 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:22 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:23 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ))) & (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set ($#k29_sin_cos :::"tan"::: ) ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR14:24 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ))) & (Bool (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR14:25 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ))) & (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set ($#k30_sin_cos :::"cot"::: ) ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR14:26 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#~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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k4_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "Z")) ($#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_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR14:27 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#~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")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k4_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "Z")) ($#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 "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR14:28 (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 ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (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 ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:29 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:30 (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 "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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (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 "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")))) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "a")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#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 :: INTEGR14:31 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "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"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "a")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (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")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "a")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "a")) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR14:32 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (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 ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:33 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (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 ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:34 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (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 "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:35 (Bool "for" (Set (Var "A")) "being" ($#~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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (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 "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:36 (Bool "for" (Set (Var "A")) "being" ($#~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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#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_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:37 (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")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ))) & (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 "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:38 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (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"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:39 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ))) & (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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:40 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (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"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ))) & (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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:41 (Bool "for" (Set (Var "A")) "being" ($#~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"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ))) & (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 "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:42 (Bool "for" (Set (Var "A")) "being" ($#~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"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ))) & (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 "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:43 (Bool "for" (Set (Var "A")) "being" ($#~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")) "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"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ))) & (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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:44 (Bool "for" (Set (Var "A")) "being" ($#~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 "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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:45 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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 "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (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 "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:46 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:47 (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 "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (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 "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:48 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:49 (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 "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ))) & (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 ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:50 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:51 (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 "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ))) & (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 "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:52 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#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"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:53 (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 "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ))) & (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 ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:54 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:55 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:56 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:57 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:58 (Bool "for" (Set (Var "A")) "being" ($#~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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#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 ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR14:59 (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 ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ))) & (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 ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ;