:: INTEGR11 semantic presentation begin theorem :: INTEGR11:1 (Bool "(" (Bool (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 4) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 4) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" ) ")" ) ; theorem :: INTEGR11:2 (Bool "(" (Bool (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 4) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 4) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" ) ")" ) ; theorem :: INTEGR11:3 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:4 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:5 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "m")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:6 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "m")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:7 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "m")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:8 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:9 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:10 (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos2 :::"cosh"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_sin_cos2 :::"sinh"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos2 :::"cosh"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_sin_cos2 :::"sinh"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos2 :::"sinh"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" ) ; theorem :: INTEGR11:11 (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos2 :::"sinh"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k4_sin_cos2 :::"cosh"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos2 :::"sinh"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k4_sin_cos2 :::"cosh"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k4_sin_cos2 :::"cosh"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" ) ; theorem :: INTEGR11:12 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set (Var "b")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set (Var "n")))) ")" ) ")" ))) ; begin theorem :: INTEGR11:13 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 4) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 4) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))) ; theorem :: INTEGR11:14 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)))) ; theorem :: INTEGR11:15 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k32_sin_cos :::"PI"::: ) ))) ; theorem :: INTEGR11:16 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 4) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 4) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))) ; theorem :: INTEGR11:17 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)))) ; theorem :: INTEGR11:18 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k32_sin_cos :::"PI"::: ) ))) ; theorem :: INTEGR11:19 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR11:20 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: INTEGR11:21 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: INTEGR11:22 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#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 "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR11:23 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: INTEGR11:24 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: INTEGR11:25 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "m")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR11:26 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "m")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR11:27 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "m")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k7_real_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "m")) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR11:28 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR11:29 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "n")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR11:30 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos2 :::"sinh"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos2 :::"cosh"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_sin_cos2 :::"sinh"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos2 :::"cosh"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_sin_cos2 :::"sinh"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))) ; theorem :: INTEGR11:31 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos2 :::"cosh"::: ) ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos2 :::"sinh"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k4_sin_cos2 :::"cosh"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos2 :::"sinh"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k4_sin_cos2 :::"cosh"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))) ; theorem :: INTEGR11:32 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; begin theorem :: INTEGR11:33 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2)))) "holds" (Bool "(" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" ))) ; theorem :: INTEGR11:34 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:35 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:36 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f1")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f2")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f1")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"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 (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f2")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f2")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:37 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:38 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"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 ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:39 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:40 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:41 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (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 (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:42 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (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 (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:43 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (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 (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:44 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (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 (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:45 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:46 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR11:47 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:48 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "r")) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"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 (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR11:49 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:50 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"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 "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_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 "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:51 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:52 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ))) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"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 "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_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 "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:53 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:54 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"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 "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k3_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 "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:55 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ))) ")" ) ")" )) ; theorem :: INTEGR11:56 (Bool "for" (Set (Var "A")) "being" ($#~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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ))) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"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 "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k47_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 "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:57 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" ) ")" )) ; theorem :: INTEGR11:58 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (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 ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:59 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" ) ")" )) ; theorem :: INTEGR11:60 (Bool "for" (Set (Var "A")) "being" ($#~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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" ) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:61 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) & (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#k29_sin_cos :::"tan"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"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 ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:62 (Bool "for" (Set (Var "A")) "being" ($#~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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_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 (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:63 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (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")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#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"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:64 (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"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (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")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:65 (Bool "for" (Set (Var "A")) "being" ($#~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 "(" "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 ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#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 "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#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 ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: INTEGR11:66 (Bool "for" (Set (Var "A")) "being" ($#~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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:67 (Bool "for" (Set (Var "A")) "being" ($#~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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:68 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")))) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:69 (Bool "for" (Set (Var "a")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")))) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k5_square_1 :::"^2"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f3")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k10_real_1 :::"/"::: ) (Set (Var "a")))) & (Bool (Set (Set (Var "f3")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "f3")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ))) & (Bool (Set (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR11:70 (Bool "for" (Set (Var "A")) "being" ($#~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 ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")))) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:71 (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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")))) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k5_square_1 :::"^2"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f3")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k10_real_1 :::"/"::: ) (Set (Var "a")))) & (Bool (Set (Set (Var "f3")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "f3")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ))) & (Bool (Set (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f3")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" ))))))) ; theorem :: INTEGR11:72 (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"::: ) ) (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ; theorem :: INTEGR11:73 (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"::: ) ) (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ")" )))) "holds" (Bool (Set ($#k2_integra5 :::"integral"::: ) "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "A")) ")" ) ")" )))))) ;