:: SIN_COS6 semantic presentation begin theorem :: SIN_COS6:1 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set (Var "r")) ($#k7_xcmplx_0 :::"/"::: ) (Set (Var "s")) ")" ) ($#k1_int_1 :::"/]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: SIN_COS6:2 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; begin theorem :: SIN_COS6:3 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))))) ; theorem :: SIN_COS6:4 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) ; theorem :: SIN_COS6:5 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))))) ; theorem :: SIN_COS6:6 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) ; registration cluster (Set ($#k31_sin_cos :::"PI"::: ) ) -> ($#v2_xxreal_0 :::"positive"::: ) ; end; theorem :: SIN_COS6:7 (Bool "(" (Bool (Set ($#k18_sin_cos :::"sin"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) ")" ) ; theorem :: SIN_COS6:8 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "r")) ($#k3_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ))))) ; theorem :: SIN_COS6:9 (Bool "(" (Bool (Set ($#k21_sin_cos :::"cos"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ; theorem :: SIN_COS6:10 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "r")) ($#k3_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ))))) ; theorem :: SIN_COS6:11 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SIN_COS6:12 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SIN_COS6:13 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SIN_COS6:14 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SIN_COS6:15 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SIN_COS6:16 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SIN_COS6:17 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SIN_COS6:18 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SIN_COS6:19 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SIN_COS6:20 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SIN_COS6:21 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ))) & (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Bool "not" (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")))))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ))))) ; theorem :: SIN_COS6:22 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ))) & (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Bool "not" (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ))))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ))))) ; theorem :: SIN_COS6:23 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set (Var "r")) ($#k6_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ")" ) ($#k1_int_1 :::"/]"::: ) ) ")" )))) ; theorem :: SIN_COS6:24 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set (Var "r")) ($#k6_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ")" ) ($#k1_int_1 :::"/]"::: ) ) ")" )))) ; theorem :: SIN_COS6:25 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set (Var "r")) ($#k6_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ")" ) ($#k1_int_1 :::"/]"::: ) ) ")" )))) ; theorem :: SIN_COS6:26 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set (Var "r")) ($#k6_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ")" ) ($#k1_int_1 :::"/]"::: ) )))) ; theorem :: SIN_COS6:27 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ))) & (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) )))) ; theorem :: SIN_COS6:28 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ))) & (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)))) ; theorem :: SIN_COS6:29 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ))) & (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k32_sin_cos :::"PI"::: ) ))) ; theorem :: SIN_COS6:30 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) ; theorem :: SIN_COS6:31 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) ; theorem :: SIN_COS6:32 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) ; theorem :: SIN_COS6:33 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) ; theorem :: SIN_COS6:34 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) ; theorem :: SIN_COS6:35 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) ; theorem :: SIN_COS6:36 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k32_sin_cos :::"PI"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) ; theorem :: SIN_COS6:37 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)))) ; theorem :: SIN_COS6:38 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))))) ; theorem :: SIN_COS6:39 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))))) ; theorem :: SIN_COS6:40 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)))) ; theorem :: SIN_COS6:41 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)))) ; theorem :: SIN_COS6:42 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))))) ; theorem :: SIN_COS6:43 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))))) ; theorem :: SIN_COS6:44 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k21_sin_cos :::"cos"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k4_numbers :::"INT"::: ) ))) ; theorem :: SIN_COS6:45 (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k7_relset_1 :::".:"::: ) (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) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) ; theorem :: SIN_COS6:46 (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) ; theorem :: SIN_COS6:47 (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) ; theorem :: SIN_COS6:48 (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) ; theorem :: SIN_COS6:49 (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) ; theorem :: SIN_COS6:50 (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) ; theorem :: SIN_COS6:51 (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) ; theorem :: SIN_COS6:52 (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) ; theorem :: SIN_COS6:53 (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) "," (Set "(" (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v5_valued_0 :::"increasing"::: ) )) ; theorem :: SIN_COS6:54 (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v6_valued_0 :::"decreasing"::: ) )) ; theorem :: SIN_COS6:55 (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v6_valued_0 :::"decreasing"::: ) )) ; theorem :: SIN_COS6:56 (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) "," (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v5_valued_0 :::"increasing"::: ) )) ; theorem :: SIN_COS6:57 (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) "," (Set "(" (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: SIN_COS6:58 (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; registration cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#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 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; registration cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) "," (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; registration cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) "," (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; theorem :: SIN_COS6:59 (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: SIN_COS6:60 (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) "," (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; registration cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; registration cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) "," (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; registration cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) "," (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; cluster (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) "," (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k2_rcomp_1 :::".["::: ) )) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; theorem :: SIN_COS6:61 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "i")) "being" ($#v1_int_1 :::"integer"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ))) & (Bool (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "i")) ")" ))) & (Bool (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k17_sin_cos :::"sin"::: ) (Set (Var "s")))) & (Bool (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k20_sin_cos :::"cos"::: ) (Set (Var "s"))))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Var "s"))))) ; begin definitionfunc :::"arcsin"::: -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: SIN_COS6:def 1 (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relset_1 :::"|"::: ) (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) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ")" ) ($#k2_partfun2 :::"""::: ) ); end; :: deftheorem defines :::"arcsin"::: SIN_COS6:def 1 : (Bool (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relset_1 :::"|"::: ) (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) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ")" ) ($#k2_partfun2 :::"""::: ) )); definitionlet "r" be ($#m1_hidden :::"set"::: ) ; func :::"arcsin"::: "r" -> ($#m1_hidden :::"set"::: ) equals :: SIN_COS6:def 2 (Set (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_seq_1 :::"."::: ) "r"); end; :: deftheorem defines :::"arcsin"::: SIN_COS6:def 2 : (Bool "for" (Set (Var "r")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_sin_cos6 :::"arcsin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))))); definitionlet "r" be ($#m1_hidden :::"set"::: ) ; :: original: :::"arcsin"::: redefine func :::"arcsin"::: "r" -> ($#m1_subset_1 :::"Real":::); end; theorem :: SIN_COS6:62 (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) )) ($#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) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) ; registration cluster (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; theorem :: SIN_COS6:63 (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) ; theorem :: SIN_COS6:64 (Bool (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#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 :::".]"::: ) ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) ))) ; theorem :: SIN_COS6:65 (Bool (Set (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k4_relset_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#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 :::".]"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) ))) ; theorem :: SIN_COS6:66 (Bool (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#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 :::".]"::: ) ) ")" ) ($#k4_relset_1 :::"*"::: ) (Set ($#k1_sin_cos6 :::"arcsin"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (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) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) ; theorem :: SIN_COS6:67 (Bool (Set (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#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 :::".]"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (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) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) ; theorem :: SIN_COS6:68 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set ($#k18_sin_cos :::"sin"::: ) (Set "(" ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "r")))) ; theorem :: SIN_COS6:69 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)))) "holds" (Bool (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set "(" ($#k17_sin_cos :::"sin"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "r")))) ; theorem :: SIN_COS6:70 (Bool (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ))) ; theorem :: SIN_COS6:71 (Bool (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ; theorem :: SIN_COS6:72 (Bool (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2))) ; theorem :: SIN_COS6:73 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" )))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) ; theorem :: SIN_COS6:74 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: SIN_COS6:75 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: SIN_COS6:76 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r")))) & (Bool (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2))) ")" )) ; theorem :: SIN_COS6:77 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r")))) & (Bool (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2))) ")" )) ; theorem :: SIN_COS6:78 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" ($#k3_sin_cos6 :::"arcsin"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) ")" )))) ; theorem :: SIN_COS6:79 (Bool "for" (Set (Var "s")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Set "(" (Set (Var "r")) ($#k3_square_1 :::"^2"::: ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "s")) ($#k3_square_1 :::"^2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k21_sin_cos :::"cos"::: ) (Set "(" ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "s")))) ; theorem :: SIN_COS6:80 (Bool "for" (Set (Var "s")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "r")) ($#k3_square_1 :::"^2"::: ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "s")) ($#k3_square_1 :::"^2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k21_sin_cos :::"cos"::: ) (Set "(" ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "s"))))) ; theorem :: SIN_COS6:81 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set ($#k21_sin_cos :::"cos"::: ) (Set "(" ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "r")) ($#k3_square_1 :::"^2"::: ) ")" ) ")" )))) ; theorem :: SIN_COS6:82 (Bool (Set (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v5_valued_0 :::"increasing"::: ) ) ; theorem :: SIN_COS6:83 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & "(" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "implies" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "r")) ($#k3_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ")" )) ; theorem :: SIN_COS6:84 (Bool (Set (Set ($#k1_sin_cos6 :::"arcsin"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v1_fcont_1 :::"continuous"::: ) ) ; begin definitionfunc :::"arccos"::: -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: SIN_COS6:def 3 (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ) ")" ) ($#k2_partfun2 :::"""::: ) ); end; :: deftheorem defines :::"arccos"::: SIN_COS6:def 3 : (Bool (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ) ")" ) ($#k2_partfun2 :::"""::: ) )); definitionlet "r" be ($#m1_hidden :::"set"::: ) ; func :::"arccos"::: "r" -> ($#m1_hidden :::"set"::: ) equals :: SIN_COS6:def 4 (Set (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_seq_1 :::"."::: ) "r"); end; :: deftheorem defines :::"arccos"::: SIN_COS6:def 4 : (Bool "for" (Set (Var "r")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k5_sin_cos6 :::"arccos"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))))); definitionlet "r" be ($#m1_hidden :::"set"::: ) ; :: original: :::"arccos"::: redefine func :::"arccos"::: "r" -> ($#m1_subset_1 :::"Real":::); end; theorem :: SIN_COS6:85 (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) )) ; registration cluster (Set ($#k4_sin_cos6 :::"arccos"::: ) ) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; theorem :: SIN_COS6:86 (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) ; theorem :: SIN_COS6:87 (Bool (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) ))) ; theorem :: SIN_COS6:88 (Bool (Set (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k4_relset_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) ))) ; theorem :: SIN_COS6:89 (Bool (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ) ")" ) ($#k4_relset_1 :::"*"::: ) (Set ($#k4_sin_cos6 :::"arccos"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ))) ; theorem :: SIN_COS6:90 (Bool (Set (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k1_rcomp_1 :::".]"::: ) ))) ; theorem :: SIN_COS6:91 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set ($#k21_sin_cos :::"cos"::: ) (Set "(" ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "r")))) ; theorem :: SIN_COS6:92 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k32_sin_cos :::"PI"::: ) ))) "holds" (Bool (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set "(" ($#k20_sin_cos :::"cos"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "r")))) ; theorem :: SIN_COS6:93 (Bool (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k32_sin_cos :::"PI"::: ) )) ; theorem :: SIN_COS6:94 (Bool (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2))) ; theorem :: SIN_COS6:95 (Bool (Set ($#k6_sin_cos6 :::"arccos"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ; theorem :: SIN_COS6:96 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: SIN_COS6:97 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: SIN_COS6:98 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set ($#k32_sin_cos :::"PI"::: ) ))) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) ; theorem :: SIN_COS6:99 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r")))) & (Bool (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k32_sin_cos :::"PI"::: ) )) ")" )) ; theorem :: SIN_COS6:100 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r")))) & (Bool (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k32_sin_cos :::"PI"::: ) )) ")" )) ; theorem :: SIN_COS6:101 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_sin_cos6 :::"arccos"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) ")" )))) ; theorem :: SIN_COS6:102 (Bool "for" (Set (Var "s")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Set "(" (Set (Var "r")) ($#k3_square_1 :::"^2"::: ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "s")) ($#k3_square_1 :::"^2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k18_sin_cos :::"sin"::: ) (Set "(" ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "s")))) ; theorem :: SIN_COS6:103 (Bool "for" (Set (Var "s")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "r")) ($#k3_square_1 :::"^2"::: ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "s")) ($#k3_square_1 :::"^2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k18_sin_cos :::"sin"::: ) (Set "(" ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "s"))))) ; theorem :: SIN_COS6:104 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set ($#k18_sin_cos :::"sin"::: ) (Set "(" ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "r")) ($#k3_square_1 :::"^2"::: ) ")" ) ")" )))) ; theorem :: SIN_COS6:105 (Bool (Set (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v6_valued_0 :::"decreasing"::: ) ) ; theorem :: SIN_COS6:106 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & "(" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "implies" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set ($#k4_sin_cos6 :::"arccos"::: ) ) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "r")) ($#k3_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ")" )) ; theorem :: SIN_COS6:107 (Bool (Set (Set ($#k4_sin_cos6 :::"arccos"::: ) ) ($#k5_relset_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v1_fcont_1 :::"continuous"::: ) ) ; theorem :: SIN_COS6:108 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set (Set "(" ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)))) ; theorem :: SIN_COS6:109 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set (Set "(" ($#k6_sin_cos6 :::"arccos"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k3_sin_cos6 :::"arcsin"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)))) ; theorem :: SIN_COS6:110 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set (Set "(" ($#k6_sin_cos6 :::"arccos"::: ) (Set (Var "r")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k3_sin_cos6 :::"arcsin"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k32_sin_cos :::"PI"::: ) ) ($#k10_real_1 :::"/"::: ) (Num 2)))) ;