:: FCONT_1 semantic presentation begin definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x0" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; pred "f" :::"is_continuous_in"::: "x0" means :: FCONT_1:def 1 (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_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 ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) "x0")) "holds" (Bool "(" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set "f" ($#k1_seq_1 :::"."::: ) "x0") ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" )); end; :: deftheorem defines :::"is_continuous_in"::: FCONT_1:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" )) ")" ))); theorem :: FCONT_1:1 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))))) ; theorem :: FCONT_1:2 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_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 ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#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")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#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")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" )) ")" ))) ; theorem :: FCONT_1:3 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x0")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" ))) ")" ))) ; theorem :: FCONT_1:4 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "for" (Set (Var "N1")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "for" (Set (Var "x1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x1"))) ($#r2_hidden :::"in"::: ) (Set (Var "N1")))))) ")" ))) ; theorem :: FCONT_1:5 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "for" (Set (Var "N1")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "N"))) ($#r1_tarski :::"c="::: ) (Set (Var "N1"))))) ")" ))) ; theorem :: FCONT_1:6 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "x0")) ($#k1_seq_4 :::"}"::: ) )))) "holds" (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: FCONT_1:7 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Var "f1")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ))) ; theorem :: FCONT_1:8 (Bool "for" (Set (Var "x0")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: FCONT_1:9 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set ($#k56_valued_1 :::"abs"::: ) (Set (Var "f"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set (Var "f"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ))) ; theorem :: FCONT_1:10 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: FCONT_1:11 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool (Set (Var "f1")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "f2")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "f2")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f1"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: FCONT_1:12 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "f1")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); attr "f" is :::"continuous"::: means :: FCONT_1:def 2 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f"))) "holds" (Bool "f" ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))); end; :: deftheorem defines :::"continuous"::: FCONT_1:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_fcont_1 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) ")" )); theorem :: FCONT_1:13 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_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" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" )) ")" ))) ; theorem :: FCONT_1:14 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "x0")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x0")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" ))) ")" ))) ; registration cluster ($#v1_funct_1 :::"Function-like"::: ) bbbadV3_FUNCT_1() -> ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_valued_0 :::"complex-valued"::: ) ($#v2_valued_0 :::"ext-real-valued"::: ) ($#v3_valued_0 :::"real-valued"::: ) ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "f" be ($#v1_fcont_1 :::"continuous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: FCONT_1:15 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) "iff" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) ")" ))) ; theorem :: FCONT_1:16 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; registration cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_xboole_0 :::"empty"::: ) -> ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "X" be ($#v1_zfmisc_1 :::"trivial"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: FCONT_1:17 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "x0")) ($#k1_seq_4 :::"}"::: ) )) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; registrationlet "f1", "f2" be ($#v1_fcont_1 :::"continuous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set "f1" ($#k1_valued_1 :::"+"::: ) "f2") -> ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set "f1" ($#k45_valued_1 :::"-"::: ) "f2") -> ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set "f1" ($#k18_valued_1 :::"(#)"::: ) "f2") -> ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: FCONT_1:18 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) ")" ))) ; theorem :: FCONT_1:19 (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v1_fcont_1 :::"continuous"::: ) ) ")" ))) ; registrationlet "f" be ($#v1_fcont_1 :::"continuous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "r" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set "r" ($#k24_valued_1 :::"(#)"::: ) "f") -> ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: FCONT_1:20 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )))) ; theorem :: FCONT_1:21 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) ")" ))) ; theorem :: FCONT_1:22 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_seq_4 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; theorem :: FCONT_1:23 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k8_relset_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_seq_4 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; theorem :: FCONT_1:24 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set (Var "f1")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_seq_4 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f2")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f1")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; registrationlet "f1", "f2" be ($#v1_fcont_1 :::"continuous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set "f1" ($#k3_relat_1 :::"(#)"::: ) "f2") -> ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: FCONT_1:25 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "f1")) ($#k7_relset_1 :::".:"::: ) (Set (Var "X")) ")" )) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; theorem :: FCONT_1:26 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set (Var "f1")) ($#k8_relset_1 :::"""::: ) (Set (Var "X1")) ")" ) ")" )) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; theorem :: FCONT_1:27 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x1")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "x2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x2")) ")" ))) ")" ) & (Bool "ex" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) "is" ($#v1_fcont_1 :::"continuous"::: ) )) ; theorem :: FCONT_1:28 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) "is" ($#v1_rcomp_1 :::"compact"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" )) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) "is" ($#v1_rcomp_1 :::"compact"::: ) )) ; theorem :: FCONT_1:29 (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_rcomp_1 :::"compact"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y"))) "is" ($#v1_rcomp_1 :::"compact"::: ) ))) ; theorem :: FCONT_1:30 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) "is" ($#v1_rcomp_1 :::"compact"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" )) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool "ex" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")) ")" ))) ")" ))) ; theorem :: FCONT_1:31 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_rcomp_1 :::"compact"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) "holds" (Bool "ex" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y")) ")" ))) ")" )))) ; definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); attr "f" is :::"Lipschitzian"::: means :: FCONT_1:def 3 (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f"))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "x1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "x2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x2")) ")" ) ")" ))) ")" ) ")" )); end; :: deftheorem defines :::"Lipschitzian"::: FCONT_1:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) "iff" (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x2")) ")" ) ")" ))) ")" ) ")" )) ")" )); theorem :: FCONT_1:32 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) "iff" (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" )))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x2")) ")" ) ")" ))) ")" ) ")" )) ")" ))) ; registration cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_xboole_0 :::"empty"::: ) -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_xboole_0 :::"empty"::: ) ($#v1_valued_0 :::"complex-valued"::: ) ($#v2_valued_0 :::"ext-real-valued"::: ) ($#v3_valued_0 :::"real-valued"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "f" be ($#v2_fcont_1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: FCONT_1:33 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ))) ; registrationlet "f1", "f2" be ($#v2_fcont_1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set "f1" ($#k1_valued_1 :::"+"::: ) "f2") -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set "f1" ($#k45_valued_1 :::"-"::: ) "f2") -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: FCONT_1:34 (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ))) ; theorem :: FCONT_1:35 (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ))) ; registrationlet "f1", "f2" be ($#v1_comseq_2 :::"bounded"::: ) ($#v2_fcont_1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set "f1" ($#k18_valued_1 :::"(#)"::: ) "f2") -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: FCONT_1:36 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "," (Set (Var "Z")) "," (Set (Var "Z1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) & (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Z"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Z1"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Z")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Z1")) ")" )) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ))) ; registrationlet "f" be ($#v2_fcont_1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "p" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set "p" ($#k24_valued_1 :::"(#)"::: ) "f") -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: FCONT_1:37 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) )))) ; registrationlet "f" be ($#v2_fcont_1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set ($#k54_valued_1 :::"|."::: ) "f" ($#k54_valued_1 :::".|"::: ) ) -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: FCONT_1:38 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) )) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" )) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) & (Bool (Set (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) ")" ))) ; registration cluster ($#v1_funct_1 :::"Function-like"::: ) bbbadV3_FUNCT_1() -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "Y" be ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set ($#k4_relat_1 :::"id"::: ) "Y") -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; registration cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v2_fcont_1 :::"Lipschitzian"::: ) -> ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: FCONT_1:39 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "r")) ($#k1_seq_4 :::"}"::: ) )))) "holds" (Bool (Set (Var "f")) "is" ($#v1_fcont_1 :::"continuous"::: ) )) ; theorem :: FCONT_1:40 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v1_fcont_1 :::"continuous"::: ) )) ; theorem :: FCONT_1:41 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "x0")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "p")))) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )))) ; theorem :: FCONT_1:42 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x0")) ($#k3_square_1 :::"^2"::: ) )) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" )) "is" ($#v1_fcont_1 :::"continuous"::: ) )) ; theorem :: FCONT_1:43 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x0")) ($#k3_square_1 :::"^2"::: ) )) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; theorem :: FCONT_1:44 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k18_complex1 :::"abs"::: ) (Set (Var "x0")))) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v1_fcont_1 :::"continuous"::: ) )) ; theorem :: FCONT_1:45 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k18_complex1 :::"abs"::: ) (Set (Var "x0")))) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; theorem :: FCONT_1:46 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_rfunct_2 :::"monotone"::: ) ) & (Bool "ex" (Set (Var "p")) "," (Set (Var "g")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "g"))) & (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k1_rcomp_1 :::".]"::: ) )) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; theorem :: FCONT_1:47 (Bool "for" (Set (Var "p")) "," (Set (Var "g")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "g"))) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v5_valued_0 :::"increasing"::: ) ) "or" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#v6_valued_0 :::"decreasing"::: ) ) ")" )) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k1_rcomp_1 :::".]"::: ) ) ")" ) ($#k2_partfun2 :::"""::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k1_rcomp_1 :::".]"::: ) ) ")" )) "is" ($#v1_fcont_1 :::"continuous"::: ) ))) ; definitionlet "a", "b" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"AffineMap"::: "(" "a" "," "b" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: FCONT_1:def 4 (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "a" ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "x")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) "b"))); end; :: deftheorem defines :::"AffineMap"::: FCONT_1:def 4 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" )) "iff" (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "x")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "b"))))) ")" ))); registrationlet "a", "b" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set ($#k1_fcont_1 :::"AffineMap"::: ) "(" "a" "," "b" ")" ) -> ($#v1_fcont_1 :::"continuous"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v1_valued_0 :::"complex-valued"::: ) ($#v2_valued_0 :::"ext-real-valued"::: ) ($#v3_valued_0 :::"real-valued"::: ) ($#v1_fcont_1 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: FCONT_1:48 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "b")))) ; theorem :: FCONT_1:49 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "b"))))) ; theorem :: FCONT_1:50 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: FCONT_1:51 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y"))))) ; theorem :: FCONT_1:52 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y"))))) ; theorem :: FCONT_1:53 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "y")))) "holds" (Bool (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y"))))) ; theorem :: FCONT_1:54 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "y")))) "holds" (Bool (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y"))))) ; theorem :: FCONT_1:55 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) ; theorem :: FCONT_1:56 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k2_funct_1 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set "(" (Set (Var "a")) ($#k5_xcmplx_0 :::"""::: ) ")" ) "," (Set "(" ($#k10_complex1 :::"-"::: ) (Set "(" (Set (Var "b")) ($#k13_complex1 :::"/"::: ) (Set (Var "a")) ")" ) ")" ) ")" ))) ; theorem :: FCONT_1:57 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_fcont_1 :::"AffineMap"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "b")) "," (Set "(" (Set (Var "a")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "b")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) ;