:: COMSEQ_1 semantic presentation begin definitionmode Complex_Sequence is ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ); end; theorem :: COMSEQ_1:1 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m1_subset_1 :::"Complex_Sequence":::)) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k5_numbers :::"NAT"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k5_numbers :::"NAT"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) )) ")" ) ")" ) ")" )) ; theorem :: COMSEQ_1:2 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m1_subset_1 :::"Complex_Sequence":::)) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k5_numbers :::"NAT"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) )) ")" ) ")" ) ")" )) ; scheme :: COMSEQ_1:sch 1 ExComplexSeq{ F1( ($#m1_hidden :::"set"::: ) ) -> ($#v1_xcmplx_0 :::"complex"::: ) ($#m1_hidden :::"number"::: ) } : (Bool "ex" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k8_nat_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set F1 "(" (Set (Var "n")) ")" )))) proof end; notationlet "f" be ($#v1_valued_0 :::"complex-valued"::: ) ($#m1_hidden :::"Relation":::); synonym :::"non-zero"::: "f" for :::"non-empty"::: ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v2_relat_1 :::"non-zero"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) bbbadV1_PARTFUN1((Set ($#k5_numbers :::"NAT"::: ) )) bbbadV1_FUNCT_2((Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) )) ($#v1_valued_0 :::"complex-valued"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ))))); end; theorem :: COMSEQ_1:3 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k5_numbers :::"NAT"::: ) ))) "holds" (Bool (Set (Set (Var "seq")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k5_complex1 :::"0c"::: ) ))) ")" )) ; theorem :: COMSEQ_1:4 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) ) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k8_nat_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k5_complex1 :::"0c"::: ) ))) ")" )) ; theorem :: COMSEQ_1:5 (Bool "for" (Set (Var "IT")) "being" ($#v2_relat_1 :::"non-zero"::: ) ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "IT"))) ($#r1_tarski :::"c="::: ) (Set (Set ($#k2_numbers :::"COMPLEX"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k5_complex1 :::"0c"::: ) ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: COMSEQ_1:6 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "ex" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: COMSEQ_1:7 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "," (Set (Var "seq3")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq3"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "seq2")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq3")) ")" )))) ; theorem :: COMSEQ_1:8 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "," (Set (Var "seq3")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq3"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq2")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq3")) ")" )))) ; theorem :: COMSEQ_1:9 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "," (Set (Var "seq3")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq3"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq3")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "seq2")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq3")) ")" )))) ; theorem :: COMSEQ_1:10 (Bool "for" (Set (Var "seq3")) "," (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "seq3")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq3")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "seq3")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" )))) ; theorem :: COMSEQ_1:11 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set ($#k31_valued_1 :::"-"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq"))))) ; theorem :: COMSEQ_1:12 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq2")))))) ; theorem :: COMSEQ_1:13 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ))))) ; theorem :: COMSEQ_1:14 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "," (Set (Var "seq3")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq2")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq3"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq3")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq2")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq3")) ")" )))) ; theorem :: COMSEQ_1:15 (Bool "for" (Set (Var "seq3")) "," (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "seq3")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq3")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "seq3")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq2")) ")" )))) ; theorem :: COMSEQ_1:16 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ))))) ; theorem :: COMSEQ_1:17 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k9_complex1 :::"*"::: ) (Set (Var "p")) ")" ) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "p")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq")) ")" ))))) ; theorem :: COMSEQ_1:18 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ))))) ; theorem :: COMSEQ_1:19 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")))))) ; theorem :: COMSEQ_1:20 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "," (Set (Var "seq3")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq2")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq3")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq2")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq3"))))) ; theorem :: COMSEQ_1:21 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set ($#k6_complex1 :::"1r"::: ) ) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Var "seq")))) ; theorem :: COMSEQ_1:22 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set ($#k31_valued_1 :::"-"::: ) (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "seq")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Var "seq")))) ; theorem :: COMSEQ_1:23 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "seq2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq2"))))) ; theorem :: COMSEQ_1:24 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "," (Set (Var "seq3")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq2")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq3")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq2")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq3"))))) ; theorem :: COMSEQ_1:25 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "," (Set (Var "seq3")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "seq2")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq3")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq3"))))) ; theorem :: COMSEQ_1:26 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool "(" (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "seq1")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq2"))) ($#r2_relset_1 :::"="::: ) (Set ($#k31_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ))) & (Bool (Set (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "seq2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set ($#k31_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ))) ")" )) ; theorem :: COMSEQ_1:27 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool (Set (Set (Var "seq")) ($#k36_valued_1 :::"""::: ) ) "is" ($#v2_relat_1 :::"non-zero"::: ) )) ; theorem :: COMSEQ_1:28 canceled; theorem :: COMSEQ_1:29 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) ) & (Bool (Set (Var "seq1")) "is" ($#v2_relat_1 :::"non-zero"::: ) ) ")" ) "iff" (Bool (Set (Set (Var "seq")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1"))) "is" ($#v2_relat_1 :::"non-zero"::: ) ) ")" )) ; theorem :: COMSEQ_1:30 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "seq")) ($#k36_valued_1 :::"""::: ) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq1")) ($#k36_valued_1 :::"""::: ) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k36_valued_1 :::"""::: ) ))) ; theorem :: COMSEQ_1:31 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Var "seq1")))) ; theorem :: COMSEQ_1:32 (Bool "for" (Set (Var "seq9")) "," (Set (Var "seq")) "," (Set (Var "seq19")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "seq9")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq19")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq1")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq9")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq19")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set "(" (Set (Var "seq")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" )))) ; theorem :: COMSEQ_1:33 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) ) & (Bool (Set (Var "seq1")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool (Set (Set (Var "seq")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq1"))) "is" ($#v2_relat_1 :::"non-zero"::: ) )) ; theorem :: COMSEQ_1:34 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "seq")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq1")) ")" ) ($#k36_valued_1 :::"""::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq"))))) ; theorem :: COMSEQ_1:35 (Bool "for" (Set (Var "seq2")) "," (Set (Var "seq1")) "," (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "seq2")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq2")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq"))))) ; theorem :: COMSEQ_1:36 (Bool "for" (Set (Var "seq2")) "," (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set (Var "seq2")) ($#k51_valued_1 :::"/""::: ) (Set "(" (Set (Var "seq")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq1")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq2")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq"))))) ; theorem :: COMSEQ_1:37 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "," (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool (Set (Set (Var "seq2")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq2")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set "(" (Set (Var "seq")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" )))) ; theorem :: COMSEQ_1:38 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k5_complex1 :::"0c"::: ) )) & (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq"))) "is" ($#v2_relat_1 :::"non-zero"::: ) ))) ; theorem :: COMSEQ_1:39 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool (Set ($#k31_valued_1 :::"-"::: ) (Set (Var "seq"))) "is" ($#v2_relat_1 :::"non-zero"::: ) )) ; theorem :: COMSEQ_1:40 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq")) ")" ) ($#k36_valued_1 :::"""::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k12_complex1 :::"""::: ) ")" ) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq")) ($#k36_valued_1 :::"""::: ) ")" ))))) ; theorem :: COMSEQ_1:41 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "seq")) ")" ) ($#k36_valued_1 :::"""::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq")) ($#k36_valued_1 :::"""::: ) ")" )))) ; theorem :: COMSEQ_1:42 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool "(" (Bool (Set ($#k31_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "seq1")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")))) & (Bool (Set (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "seq")) ")" )) ($#r2_relset_1 :::"="::: ) (Set ($#k31_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" ))) ")" )) ; theorem :: COMSEQ_1:43 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq")) "," (Set (Var "seq19")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "seq19")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq19")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")))) & (Bool (Set (Set "(" (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq19")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq19")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")))) ")" )) ; theorem :: COMSEQ_1:44 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq9")) "," (Set (Var "seq1")) "," (Set (Var "seq19")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) ) & (Bool (Set (Var "seq9")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "seq19")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq9")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq9")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "seq19")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq")) ")" ) ")" ) ($#k51_valued_1 :::"/""::: ) (Set "(" (Set (Var "seq")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq9")) ")" ))) & (Bool (Set (Set "(" (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq19")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq9")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq9")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq19")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq")) ")" ) ")" ) ($#k51_valued_1 :::"/""::: ) (Set "(" (Set (Var "seq")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq9")) ")" ))) ")" )) ; theorem :: COMSEQ_1:45 (Bool "for" (Set (Var "seq19")) "," (Set (Var "seq")) "," (Set (Var "seq9")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "seq19")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set "(" (Set (Var "seq9")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq1")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq19")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set "(" (Set (Var "seq")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq9")) ")" )))) ; theorem :: COMSEQ_1:46 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq9")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set ($#k55_valued_1 :::"|."::: ) (Set "(" (Set (Var "seq")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq9")) ")" ) ($#k55_valued_1 :::".|"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "seq")) ($#k55_valued_1 :::".|"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "seq9")) ($#k55_valued_1 :::".|"::: ) )))) ; theorem :: COMSEQ_1:47 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "seq")) ($#k55_valued_1 :::".|"::: ) ) "is" ($#v2_relat_1 :::"non-zero"::: ) )) ; theorem :: COMSEQ_1:48 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "seq")) ($#k55_valued_1 :::".|"::: ) ) ($#k37_valued_1 :::"""::: ) ) ($#r2_relset_1 :::"="::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set "(" (Set (Var "seq")) ($#k36_valued_1 :::"""::: ) ")" ) ($#k55_valued_1 :::".|"::: ) ))) ; theorem :: COMSEQ_1:49 (Bool "for" (Set (Var "seq9")) "," (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set ($#k55_valued_1 :::"|."::: ) (Set "(" (Set (Var "seq9")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" ) ($#k55_valued_1 :::".|"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "seq9")) ($#k55_valued_1 :::".|"::: ) ) ($#k52_valued_1 :::"/""::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "seq")) ($#k55_valued_1 :::".|"::: ) )))) ; theorem :: COMSEQ_1:50 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool (Set ($#k55_valued_1 :::"|."::: ) (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq")) ")" ) ($#k55_valued_1 :::".|"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k17_complex1 :::"|."::: ) (Set (Var "r")) ($#k17_complex1 :::".|"::: ) ) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "seq")) ($#k55_valued_1 :::".|"::: ) ))))) ;