:: IRRAT_1 semantic presentation begin notationlet "x" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; antonym :::"irrational"::: "x" for :::"rational"::: ; end; notationlet "x", "y" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; synonym "x" :::"^"::: "y" for "x" :::"to_power"::: "y"; end; theorem :: IRRAT_1:1 (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v1_int_2 :::"prime"::: ) )) "holds" (Bool (Set ($#k7_square_1 :::"sqrt"::: ) (Set (Var "p"))) "is" ($#v1_rat_1 :::"irrational"::: ) )) ; theorem :: IRRAT_1:2 (Bool "ex" (Set (Var "x")) "," (Set (Var "y")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_rat_1 :::"irrational"::: ) ) & (Bool (Set (Var "y")) "is" ($#v1_rat_1 :::"irrational"::: ) ) & (Bool (Set (Set (Var "x")) ($#k3_power :::"^"::: ) (Set (Var "y"))) "is" ($#v1_rat_1 :::"rational"::: ) ) ")" )) ; begin scheme :: IRRAT_1:sch 1 LambdaRealSeq{ F1( ($#m1_hidden :::"set"::: ) ) -> ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) } : (Bool "(" (Bool "ex" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set F1 "(" (Set (Var "n")) ")" )))) & (Bool "(" "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set F1 "(" (Set (Var "n")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set F1 "(" (Set (Var "n")) ")" )) ")" )) "holds" (Bool (Set (Var "seq1")) ($#r1_hidden :::"="::: ) (Set (Var "seq2"))) ")" ) ")" ) proof end; definitionlet "k" be ($#m1_hidden :::"Nat":::); func :::"aseq"::: "k" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: IRRAT_1:def 1 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) "k" ")" ) ($#k13_complex1 :::"/"::: ) (Set (Var "n"))))); func :::"bseq"::: "k" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: IRRAT_1:def 2 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k6_newton :::"choose"::: ) "k" ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k3_power :::"^"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) "k" ")" ) ")" )))); end; :: deftheorem defines :::"aseq"::: IRRAT_1:def 1 : (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_irrat_1 :::"aseq"::: ) (Set (Var "k")))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ) ($#k13_complex1 :::"/"::: ) (Set (Var "n"))))) ")" ))); :: deftheorem defines :::"bseq"::: IRRAT_1:def 2 : (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_irrat_1 :::"bseq"::: ) (Set (Var "k")))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Set (Var "k")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k3_power :::"^"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ) ")" )))) ")" ))); definitionlet "n" be ($#m1_hidden :::"Nat":::); func :::"cseq"::: "n" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: IRRAT_1:def 3 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "n" ($#k6_newton :::"choose"::: ) (Set (Var "k")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" "n" ($#k3_power :::"^"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ) ")" )))); end; :: deftheorem defines :::"cseq"::: IRRAT_1:def 3 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_irrat_1 :::"cseq"::: ) (Set (Var "n")))) "iff" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b2")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Set (Var "k")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k3_power :::"^"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ) ")" )))) ")" ))); theorem :: IRRAT_1:3 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_irrat_1 :::"cseq"::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_irrat_1 :::"bseq"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))))) ; definitionfunc :::"dseq"::: -> ($#m1_subset_1 :::"Real_Sequence":::) means :: IRRAT_1:def 4 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Set (Var "n")) ")" ) ")" ) ($#k3_power :::"^"::: ) (Set (Var "n"))))); end; :: deftheorem defines :::"dseq"::: IRRAT_1:def 4 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k4_irrat_1 :::"dseq"::: ) )) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Set (Var "n")) ")" ) ")" ) ($#k3_power :::"^"::: ) (Set (Var "n"))))) ")" )); definitionfunc :::"eseq"::: -> ($#m1_subset_1 :::"Real_Sequence":::) means :: IRRAT_1:def 5 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k9_newton :::"!"::: ) ")" )))); end; :: deftheorem defines :::"eseq"::: IRRAT_1:def 5 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k5_irrat_1 :::"eseq"::: ) )) "iff" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k9_newton :::"!"::: ) ")" )))) ")" )); theorem :: IRRAT_1:4 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "n")) ($#k3_power :::"^"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k3_power :::"^"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Set (Var "n"))))) ; theorem :: IRRAT_1:5 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Set (Var "k")) ")" )))) ; theorem :: IRRAT_1:6 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k2_irrat_1 :::"bseq"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set "(" ($#k2_irrat_1 :::"bseq"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set "(" ($#k1_irrat_1 :::"aseq"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" )))) ; theorem :: IRRAT_1:7 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_irrat_1 :::"aseq"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "k")) ($#k13_complex1 :::"/"::: ) (Set (Var "n")) ")" )))) ; theorem :: IRRAT_1:8 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_irrat_1 :::"aseq"::: ) (Set (Var "k"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k1_irrat_1 :::"aseq"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) ; theorem :: IRRAT_1:9 (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" )) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ))) ; theorem :: IRRAT_1:10 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k2_irrat_1 :::"bseq"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: IRRAT_1:11 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k9_complex1 :::"*"::: ) (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k9_newton :::"!"::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k9_newton :::"!"::: ) ")" )))) ; theorem :: IRRAT_1:12 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k2_irrat_1 :::"bseq"::: ) (Set (Var "k"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k2_irrat_1 :::"bseq"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k9_newton :::"!"::: ) ")" ))) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k2_irrat_1 :::"bseq"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k5_irrat_1 :::"eseq"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "k")))) ")" )) ; theorem :: IRRAT_1:13 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k1_irrat_1 :::"aseq"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set "(" ($#k1_irrat_1 :::"aseq"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" )) ; theorem :: IRRAT_1:14 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k2_irrat_1 :::"bseq"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set "(" ($#k2_irrat_1 :::"bseq"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k9_newton :::"!"::: ) ")" ))) & (Bool (Set (Set "(" ($#k2_irrat_1 :::"bseq"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k5_irrat_1 :::"eseq"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "k")))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_irrat_1 :::"cseq"::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "k")))) & (Bool (Set (Set "(" ($#k3_irrat_1 :::"cseq"::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k9_newton :::"!"::: ) ")" ))) & (Bool (Set (Set "(" ($#k3_irrat_1 :::"cseq"::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k5_irrat_1 :::"eseq"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "k")))) ")" )) ; theorem :: IRRAT_1:15 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Num 1)) "is" ($#v1_series_1 :::"summable"::: ) )) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k4_series_1 :::"Sum"::: ) (Set "(" (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Num 1) ")" ) ")" ))) ")" )) ; theorem :: IRRAT_1:16 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "sq")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "sq"))))) "holds" (Bool (Set (Set "(" (Set (Var "sq")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "sq")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))))) ; theorem :: IRRAT_1:17 (Bool "for" (Set (Var "sq")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "sq"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "sq"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sq")) ($#k1_seq_1 :::"."::: ) (Num 1) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" (Set (Var "sq")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" ) ")" )))) ; theorem :: IRRAT_1:18 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "sq")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "sq"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "sq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "sq")))) ")" )))) ; theorem :: IRRAT_1:19 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" "(" (Set (Var "x")) "," (Set (Var "y")) ")" ($#k7_newton :::"In_Power"::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Set (Var "k")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k3_power :::"^"::: ) (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ) ")" ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k3_power :::"^"::: ) (Set (Var "k")) ")" ))))) ; theorem :: IRRAT_1:20 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k3_irrat_1 :::"cseq"::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "(" (Num 1) "," (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Set (Var "n")) ")" ) ")" ($#k7_newton :::"In_Power"::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: IRRAT_1:21 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k3_irrat_1 :::"cseq"::: ) (Set (Var "n"))) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set "(" ($#k3_irrat_1 :::"cseq"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Set (Var "n")) ")" ) ")" ) ($#k3_power :::"^"::: ) (Set (Var "n")))) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set "(" ($#k3_irrat_1 :::"cseq"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_irrat_1 :::"dseq"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" )) ; theorem :: IRRAT_1:22 (Bool "(" (Bool (Set ($#k4_irrat_1 :::"dseq"::: ) ) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set ($#k4_irrat_1 :::"dseq"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k8_power :::"number_e"::: ) )) ")" ) ; theorem :: IRRAT_1:23 (Bool "(" (Bool (Set ($#k5_irrat_1 :::"eseq"::: ) ) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set ($#k5_irrat_1 :::"eseq"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k26_sin_cos :::"exp_R"::: ) (Num 1))) ")" ) ; theorem :: IRRAT_1:24 (Bool "for" (Set (Var "K")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "dseqK")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "dseqK")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set "(" ($#k3_irrat_1 :::"cseq"::: ) (Set (Var "n")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "K")))) ")" )) "holds" (Bool "(" (Bool (Set (Var "dseqK")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "dseqK"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set ($#k5_irrat_1 :::"eseq"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "K")))) ")" ))) ; theorem :: IRRAT_1:25 (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x")))) "holds" (Bool "for" (Set (Var "eps")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "eps")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "x")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "eps"))))))))) ; theorem :: IRRAT_1:26 (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "eps")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "eps")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "x")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "eps")))))) ")" ) & (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "x")))))) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ))) ; theorem :: IRRAT_1:27 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_series_1 :::"summable"::: ) )) "holds" (Bool "for" (Set (Var "eps")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "eps")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "K")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "K"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set "(" ($#k4_series_1 :::"Sum"::: ) (Set (Var "seq")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "eps"))))))) ; theorem :: IRRAT_1:28 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set ($#k4_irrat_1 :::"dseq"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_series_1 :::"Sum"::: ) (Set ($#k5_irrat_1 :::"eseq"::: ) )))) ; theorem :: IRRAT_1:29 (Bool "for" (Set (Var "K")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set (Var "seq"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "K")))))) ; theorem :: IRRAT_1:30 (Bool "(" (Bool (Set ($#k4_irrat_1 :::"dseq"::: ) ) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set ($#k4_irrat_1 :::"dseq"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_series_1 :::"Sum"::: ) (Set ($#k5_irrat_1 :::"eseq"::: ) ))) ")" ) ; definitionredefine func :::"number_e"::: equals :: IRRAT_1:def 6 (Set ($#k4_series_1 :::"Sum"::: ) (Set ($#k5_irrat_1 :::"eseq"::: ) )); end; :: deftheorem defines :::"number_e"::: IRRAT_1:def 6 : (Bool (Set ($#k7_power :::"number_e"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_series_1 :::"Sum"::: ) (Set ($#k5_irrat_1 :::"eseq"::: ) ))); definitionredefine func :::"number_e"::: equals :: IRRAT_1:def 7 (Set ($#k26_sin_cos :::"exp_R"::: ) (Num 1)); end; :: deftheorem defines :::"number_e"::: IRRAT_1:def 7 : (Bool (Set ($#k7_power :::"number_e"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k26_sin_cos :::"exp_R"::: ) (Num 1))); begin theorem :: IRRAT_1:31 (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_rat_1 :::"rational"::: ) )) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "x"))) "is" ($#v1_int_1 :::"integer"::: ) ) ")" ))) ; theorem :: IRRAT_1:32 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set ($#k5_irrat_1 :::"eseq"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k9_newton :::"!"::: ) ")" )))) ; theorem :: IRRAT_1:33 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k9_newton :::"!"::: ) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: IRRAT_1:34 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set (Var "seq"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: IRRAT_1:35 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" ($#k4_series_1 :::"Sum"::: ) (Set "(" (Set ($#k5_irrat_1 :::"eseq"::: ) ) ($#k1_valued_0 :::"^\"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: IRRAT_1:36 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k9_newton :::"!"::: ) ")" )) "is" ($#v1_int_1 :::"integer"::: ) )) ; theorem :: IRRAT_1:37 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set ($#k5_irrat_1 :::"eseq"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" )) "is" ($#v1_int_1 :::"integer"::: ) )) ; theorem :: IRRAT_1:38 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k9_newton :::"!"::: ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "x")) ($#k3_power :::"^"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))) ; theorem :: IRRAT_1:39 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" ($#k4_series_1 :::"Sum"::: ) (Set "(" (Set ($#k5_irrat_1 :::"eseq"::: ) ) ($#k1_valued_0 :::"^\"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "x")) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x")) ")" ))))) ; theorem :: IRRAT_1:40 (Bool "for" (Set (Var "x")) "," (Set (Var "n")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k13_complex1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool (Set (Set (Var "x")) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) ; theorem :: IRRAT_1:41 (Bool (Set ($#k7_power :::"number_e"::: ) ) "is" ($#v1_rat_1 :::"irrational"::: ) ) ;