:: CFCONT_1 semantic presentation begin definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); let "x0" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ); pred "f" :::"is_continuous_in"::: "x0" means :: CFCONT_1:def 1 (Bool "(" (Bool "x0" ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) "x0")) "holds" (Bool "(" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set "f" ($#k7_partfun1 :::"/."::: ) "x0") ($#r1_hidden :::"="::: ) (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" ) ")" ) ")" ); end; :: deftheorem defines :::"is_continuous_in"::: CFCONT_1:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" ) ")" ) ")" ) ")" ))); theorem :: CFCONT_1:1 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "," (Set (Var "seq3")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "seq1")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "seq2")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq3")))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "seq2")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" (Set (Var "seq3")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" )))) ")" )) ; theorem :: CFCONT_1:2 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "Ns")) "being" bbbadV5_VALUED_0() ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" ) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "seq2")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ))) & (Bool (Set (Set "(" (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq2")) ")" ) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq2")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ))) & (Bool (Set (Set "(" (Set (Var "seq1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq2")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ))) ")" ))) ; theorem :: CFCONT_1:3 (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "Ns")) "being" bbbadV5_VALUED_0() ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq")) ")" ) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "g")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "seq")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" )))))) ; theorem :: CFCONT_1:4 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "Ns")) "being" bbbadV5_VALUED_0() ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "seq")) ")" ) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns"))) ($#r2_relset_1 :::"="::: ) (Set ($#k31_valued_1 :::"-"::: ) (Set "(" (Set (Var "seq")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ))) & (Bool (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "seq")) ($#k55_valued_1 :::".|"::: ) ) ($#k2_valued_0 :::"*"::: ) (Set (Var "Ns"))) ($#r2_relset_1 :::"="::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set "(" (Set (Var "seq")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ) ($#k55_valued_1 :::".|"::: ) )) ")" ))) ; theorem :: CFCONT_1:5 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "Ns")) "being" bbbadV5_VALUED_0() ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "seq")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ) ($#k36_valued_1 :::"""::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq")) ($#k36_valued_1 :::"""::: ) ")" ) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")))))) ; theorem :: CFCONT_1:6 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "Ns")) "being" bbbadV5_VALUED_0() ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "seq1")) ($#k51_valued_1 :::"/""::: ) (Set (Var "seq")) ")" ) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "seq1")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ) ($#k51_valued_1 :::"/""::: ) (Set "(" (Set (Var "seq")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ))))) ; theorem :: CFCONT_1:7 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "h1")) "," (Set (Var "h2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "h1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "h2")) ")" )))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "h1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "h2")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "h1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "h2")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))) & (Bool (Set (Set "(" (Set (Var "h1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "h2")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "h1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "h2")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))) & (Bool (Set (Set "(" (Set (Var "h1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "h2")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "h1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "h2")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))) ")" ))) ; theorem :: CFCONT_1:8 (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "h")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "g")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )))))) ; theorem :: CFCONT_1:9 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set ($#k31_valued_1 :::"-"::: ) (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "h")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")))))) ; theorem :: CFCONT_1:10 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "h")) ($#k2_cfunct_1 :::"^"::: ) ")" )))) "holds" (Bool (Set (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_relat_1 :::"non-zero"::: ) ))) ; theorem :: CFCONT_1:11 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "h")) ($#k2_cfunct_1 :::"^"::: ) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k36_valued_1 :::"""::: ) )))) ; theorem :: CFCONT_1:12 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "Ns")) "being" bbbadV5_VALUED_0() ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set ($#k7_comseq_3 :::"Re"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" )) ($#r2_relset_1 :::"="::: ) (Set ($#k7_comseq_3 :::"Re"::: ) (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "seq")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ) ")" )))))) ; theorem :: CFCONT_1:13 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "Ns")) "being" bbbadV5_VALUED_0() ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set ($#k8_comseq_3 :::"Im"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" )) ($#r2_relset_1 :::"="::: ) (Set ($#k8_comseq_3 :::"Im"::: ) (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "seq")) ($#k9_comseq_3 :::"*"::: ) (Set (Var "Ns")) ")" ) ")" )))))) ; theorem :: CFCONT_1:14 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "h1")) "," (Set (Var "h2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "h1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "h2")) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "h1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "h2")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "h1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "h2")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))) & (Bool (Set (Set "(" (Set (Var "h1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "h2")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "h1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "h2")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))) & (Bool (Set (Set "(" (Set (Var "h1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "h2")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "h1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "h2")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))) ")" ))) ; theorem :: CFCONT_1:15 (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "h")) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "h")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "g")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )))))) ; theorem :: CFCONT_1:16 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "h")) ($#k5_relset_1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set (Var "h")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k5_relset_1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k36_valued_1 :::"""::: ) ))))) ; theorem :: CFCONT_1:17 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "seq"))) & (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) "holds" (Bool (Set (Var "seq1")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) ; theorem :: CFCONT_1:18 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "seq"))) & (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) "holds" (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq"))))) ; theorem :: CFCONT_1:19 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool "ex" (Set (Var "k")) "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 "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "seq1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "seq")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))))))) "holds" (Bool (Set (Var "seq1")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) ; theorem :: CFCONT_1:20 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "seq1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "seq")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))))))) "holds" (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq1"))))) ; theorem :: CFCONT_1:21 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "k"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq")))) ")" ))) ; theorem :: CFCONT_1:22 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Var "seq")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "seq1")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "k")))))) "holds" (Bool (Set (Var "seq1")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) ; theorem :: CFCONT_1:23 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Var "seq")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "seq1")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "k")))))) "holds" (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq"))))) ; theorem :: CFCONT_1:24 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "k"))) "is" ($#v2_relat_1 :::"non-zero"::: ) ))) ; theorem :: CFCONT_1:25 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool "(" (Bool (Set (Var "seq1")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "seq"))) & (Bool (Set (Var "seq1")) "is" ($#v2_relat_1 :::"non-zero"::: ) ) ")" ))) ; theorem :: CFCONT_1:26 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) ; theorem :: CFCONT_1:27 (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool "(" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq")))) ")" ) "or" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "seq")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "g")))) ")" ) ")" )) "holds" (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "g"))))) ; theorem :: CFCONT_1:28 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "seq")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))))) ; theorem :: CFCONT_1:29 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "for" (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "seq"))) & (Bool (Set (Var "seq1")) "is" ($#v2_relat_1 :::"non-zero"::: ) )) "holds" (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" (Set (Var "seq1")) ($#k36_valued_1 :::"""::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq")) ")" ) ($#k12_complex1 :::"""::: ) )))) ; theorem :: CFCONT_1:30 (Bool "for" (Set (Var "seq")) "," (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool (Set (Var "seq1")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) "holds" (Bool "(" (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" (Set (Var "seq")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "seq")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#k8_complex1 :::"+"::: ) (Set "(" ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq1")) ")" ))) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" (Set (Var "seq")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "seq")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq1")) ")" ))) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" (Set (Var "seq1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq1")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" (Set (Var "seq")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ))) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" (Set (Var "seq")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "seq1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "seq")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#k9_complex1 :::"*"::: ) (Set "(" ($#k3_comseq_2 :::"lim"::: ) (Set (Var "seq1")) ")" ))) ")" )) ; scheme :: CFCONT_1:sch 1 CompSeqChoice{ P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool P1[(Set (Var "n")) "," (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))]))) provided (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool P1[(Set (Var "n")) "," (Set (Var "g"))]))) proof end; begin theorem :: CFCONT_1:31 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set (Var "x0"))) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" ) ")" ) ")" ) ")" ))) ; theorem :: CFCONT_1:32 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set (Var "x1")) ($#k11_complex1 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" ))) ; theorem :: CFCONT_1:33 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ))) ; theorem :: CFCONT_1:34 (Bool "for" (Set (Var "x0")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "g")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: CFCONT_1:35 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set ($#k31_valued_1 :::"-"::: ) (Set (Var "f"))) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: CFCONT_1:36 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: CFCONT_1:37 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "f2")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f1"))) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_continuous_on"::: "X" means :: CFCONT_1:def 2 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set "f" ($#k5_relset_1 :::"|"::: ) "X") ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ); end; :: deftheorem defines :::"is_continuous_on"::: CFCONT_1:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k5_relset_1 :::"|"::: ) (Set (Var "X"))) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ) ")" ))); theorem :: CFCONT_1:38 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_comseq_2 :::"lim"::: ) (Set (Var "s1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" ) ")" ) ")" ) ")" ))) ; theorem :: CFCONT_1:39 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set (Var "x1")) ($#k11_complex1 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" ))) ")" ) ")" ) ")" ))) ; theorem :: CFCONT_1:40 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool (Set (Set (Var "f")) ($#k5_relset_1 :::"|"::: ) (Set (Var "X"))) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" ))) ; theorem :: CFCONT_1:41 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X1"))))) ; theorem :: CFCONT_1:42 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: CFCONT_1:43 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" ))) ; theorem :: CFCONT_1:44 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X1")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")))) & (Bool (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")))) & (Bool (Set (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")))) ")" ))) ; theorem :: CFCONT_1:45 (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "g")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: CFCONT_1:46 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k31_valued_1 :::"-"::: ) (Set (Var "f"))) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))))) ; theorem :: CFCONT_1:47 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))))) ; theorem :: CFCONT_1:48 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k5_relset_1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))))) ; theorem :: CFCONT_1:49 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "f1")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "f2")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f1"))) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))))) ; theorem :: CFCONT_1:50 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "x1")) ($#k8_complex1 :::"+"::: ) (Set (Var "x2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k8_complex1 :::"+"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ))) ")" ) & (Bool "ex" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Set (Var "f")) ($#r1_cfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) "holds" (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ))) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is :::"compact"::: means :: CFCONT_1:def 3 (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) "X")) "holds" (Bool "ex" (Set (Var "s2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s2")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "s1"))) & (Bool (Set (Var "s2")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "s2"))) ($#r2_hidden :::"in"::: ) "X") ")" ))); end; :: deftheorem defines :::"compact"::: CFCONT_1:def 3 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_cfcont_1 :::"compact"::: ) ) "iff" (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool "ex" (Set (Var "s2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s2")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "s1"))) & (Bool (Set (Var "s2")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set (Var "s2"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" ))) ")" )); theorem :: CFCONT_1:51 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) "is" ($#v1_cfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) "is" ($#v1_cfcont_1 :::"compact"::: ) )) ; theorem :: CFCONT_1:52 (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_cfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r2_cfcont_1 :::"is_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y"))) "is" ($#v1_cfcont_1 :::"compact"::: ) ))) ;