:: RSSPACE4 semantic presentation begin definitionfunc :::"the_set_of_BoundedRealSequences"::: -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) means :: RSSPACE4:def 1 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rsspace :::"the_set_of_RealSequences"::: ) )) & (Bool (Set ($#k2_rsspace :::"seq_id"::: ) (Set (Var "x"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) ")" )); end; :: deftheorem defines :::"the_set_of_BoundedRealSequences"::: RSSPACE4:def 1 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rsspace :::"the_set_of_RealSequences"::: ) )) & (Bool (Set ($#k2_rsspace :::"seq_id"::: ) (Set (Var "x"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) ")" )) ")" )); registration cluster (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) -> ($#v1_rlsub_1 :::"linearly-closed"::: ) ; end; registration cluster (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "#)" ) -> ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ; end; definitionfunc :::"linfty_norm"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: RSSPACE4:def 2 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set "(" ($#k2_rsspace :::"seq_id"::: ) (Set (Var "x")) ")" ) ")" ) ")" )))); end; :: deftheorem defines :::"linfty_norm"::: RSSPACE4:def 2 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k2_rsspace4 :::"linfty_norm"::: ) )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ))) "holds" (Bool (Set (Set (Var "b1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set "(" ($#k2_rsspace :::"seq_id"::: ) (Set (Var "x")) ")" ) ")" ) ")" )))) ")" )); theorem :: RSSPACE4:1 (Bool "for" (Set (Var "rseq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "rseq")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "rseq")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "rseq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )) ; registration cluster (Set ($#g1_normsp_1 :::"NORMSTR"::: ) "(#" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set ($#k2_rsspace4 :::"linfty_norm"::: ) ) "#)" ) -> ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ; end; definitionfunc :::"linfty_Space"::: -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_normsp_1 :::"NORMSTR"::: ) equals :: RSSPACE4:def 3 (Set ($#g1_normsp_1 :::"NORMSTR"::: ) "(#" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set ($#k2_rsspace4 :::"linfty_norm"::: ) ) "#)" ); end; :: deftheorem defines :::"linfty_Space"::: RSSPACE4:def 3 : (Bool (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#g1_normsp_1 :::"NORMSTR"::: ) "(#" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) ) "," (Set ($#k7_rsspace :::"Linear_Space_of_RealSequences"::: ) ) ")" ")" ) "," (Set ($#k2_rsspace4 :::"linfty_norm"::: ) ) "#)" )); theorem :: RSSPACE4:2 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rsspace4 :::"the_set_of_BoundedRealSequences"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) )) "iff" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Real_Sequence":::)) & (Bool (Set ($#k2_rsspace :::"seq_id"::: ) (Set (Var "x"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) ")" ) ")" ) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k3_rsspace4 :::"linfty_Space"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_rsspace :::"Zeroseq"::: ) )) & (Bool "(" "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) "holds" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k2_rsspace :::"seq_id"::: ) (Set (Var "u")))) ")" ) & (Bool "(" "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) "holds" (Bool (Set (Set (Var "u")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_rsspace :::"seq_id"::: ) (Set (Var "u")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k2_rsspace :::"seq_id"::: ) (Set (Var "v")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) "holds" (Bool (Set (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k2_rsspace :::"seq_id"::: ) (Set (Var "u")) ")" )))) ")" ) & (Bool "(" "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) "holds" (Bool "(" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k2_rsspace :::"seq_id"::: ) (Set (Var "u")) ")" ))) & (Bool (Set ($#k2_rsspace :::"seq_id"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "u")) ")" )) ($#r2_funct_2 :::"="::: ) (Set ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k2_rsspace :::"seq_id"::: ) (Set (Var "u")) ")" ))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) "holds" (Bool (Set (Set (Var "u")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_rsspace :::"seq_id"::: ) (Set (Var "u")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k2_rsspace :::"seq_id"::: ) (Set (Var "v")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) "holds" (Bool (Set ($#k2_rsspace :::"seq_id"::: ) (Set (Var "v"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) & (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "v")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set "(" ($#k2_rsspace :::"seq_id"::: ) (Set (Var "v")) ")" ) ")" ) ")" ))) ")" ) ")" ) ; theorem :: RSSPACE4:3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ))) ")" & "(" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k3_rsspace4 :::"linfty_Space"::: ) )))) "implies" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) )) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) ) ($#k7_real_1 :::"+"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "y")) ($#k1_normsp_0 :::".||"::: ) ))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set (Var "a")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) ))) ")" ))) ; registration cluster (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#v3_normsp_0 :::"discerning"::: ) ($#v4_normsp_0 :::"reflexive"::: ) ($#v2_normsp_1 :::"RealNormSpace-like"::: ) ; end; theorem :: RSSPACE4:4 (Bool "for" (Set (Var "vseq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k3_rsspace4 :::"linfty_Space"::: ) ) "st" (Bool (Bool (Set (Var "vseq")) "is" ($#v1_rsspace3 :::"Cauchy_sequence_by_Norm"::: ) )) "holds" (Bool (Set (Var "vseq")) "is" ($#v3_normsp_1 :::"convergent"::: ) )) ; begin definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); let "IT" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "Y"))); attr "IT" is :::"bounded"::: means :: RSSPACE4:def 4 (Bool "ex" (Set (Var "K")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "K"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" "IT" ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "K"))) ")" ) ")" )); end; :: deftheorem defines :::"bounded"::: RSSPACE4:def 4 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "IT")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_rsspace4 :::"bounded"::: ) ) "iff" (Bool "ex" (Set (Var "K")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "K"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "IT")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "K"))) ")" ) ")" )) ")" )))); theorem :: RSSPACE4:5 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "Y")))) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v1_rsspace4 :::"bounded"::: ) )))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) "X" ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("X") ($#v1_funct_2 :::"quasi_total"::: ) ($#v1_rsspace4 :::"bounded"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); func :::"BoundedFunctions"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) means :: RSSPACE4:def 5 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) "is" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")) ")" )); end; :: deftheorem defines :::"BoundedFunctions"::: RSSPACE4:def 5 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool (Set (Var "x")) "is" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y")))) ")" )) ")" )))); registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster (Set ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: RSSPACE4:6 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) "holds" (Bool (Set ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) ))) ; theorem :: RSSPACE4:7 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) "holds" (Bool (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "#)" ) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" )))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "#)" ) -> ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ; end; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); func :::"R_VectorSpace_of_BoundedFunctions"::: "(" "X" "," "Y" ")" -> ($#l1_rlvect_1 :::"RealLinearSpace":::) equals :: RSSPACE4:def 6 (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "#)" ); end; :: deftheorem defines :::"R_VectorSpace_of_BoundedFunctions"::: RSSPACE4:def 6 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) "holds" (Bool (Set ($#k5_rsspace4 :::"R_VectorSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "#)" )))); registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster (Set ($#k5_rsspace4 :::"R_VectorSpace_of_BoundedFunctions"::: ) "(" "X" "," "Y" ")" ) -> ($#v1_rlvect_1 :::"strict"::: ) ; end; theorem :: RSSPACE4:8 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k5_rsspace4 :::"R_VectorSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "f9")) "," (Set (Var "g9")) "," (Set (Var "h9")) "being" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "st" (Bool (Bool (Set (Var "f9")) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "g9")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "h9")) ($#r1_hidden :::"="::: ) (Set (Var "h")))) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "g")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "g9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))))) ; theorem :: RSSPACE4:9 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k5_rsspace4 :::"R_VectorSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "f9")) "," (Set (Var "h9")) "being" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "st" (Bool (Bool (Set (Var "f9")) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "h9")) ($#r1_hidden :::"="::: ) (Set (Var "h")))) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set (Var "f9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))))) ; theorem :: RSSPACE4:10 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k5_rsspace4 :::"R_VectorSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "Y")) ")" ))))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" )) ; func :::"modetrans"::: "(" "f" "," "X" "," "Y" ")" -> ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") equals :: RSSPACE4:def 7 "f"; end; :: deftheorem defines :::"modetrans"::: RSSPACE4:def 7 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ))) "holds" (Bool (Set ($#k6_rsspace4 :::"modetrans"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f")))))); definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); let "u" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "Y"))); func :::"PreNorms"::: "u" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: RSSPACE4:def 8 "{" (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" "u" ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) where t "is" ($#m1_subset_1 :::"Element"::: ) "of" "X" : (Bool verum) "}" ; end; :: deftheorem defines :::"PreNorms"::: RSSPACE4:def 8 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "holds" (Bool (Set ($#k7_rsspace4 :::"PreNorms"::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) "{" (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "u")) ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) where t "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) : (Bool verum) "}" )))); theorem :: RSSPACE4:11 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "g")) "being" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "holds" (Bool (Set ($#k7_rsspace4 :::"PreNorms"::: ) (Set (Var "g"))) "is" ($#v4_xxreal_2 :::"bounded_above"::: ) )))) ; theorem :: RSSPACE4:12 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v1_rsspace4 :::"bounded"::: ) ) "iff" (Bool (Set ($#k7_rsspace4 :::"PreNorms"::: ) (Set (Var "g"))) "is" ($#v4_xxreal_2 :::"bounded_above"::: ) ) ")" )))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); func :::"BoundedFunctionsNorm"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: RSSPACE4:def 9 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k7_rsspace4 :::"PreNorms"::: ) (Set "(" ($#k6_rsspace4 :::"modetrans"::: ) "(" (Set (Var "x")) "," "X" "," "Y" ")" ")" ) ")" )))); end; :: deftheorem defines :::"BoundedFunctionsNorm"::: RSSPACE4:def 9 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k8_rsspace4 :::"BoundedFunctionsNorm"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k7_rsspace4 :::"PreNorms"::: ) (Set "(" ($#k6_rsspace4 :::"modetrans"::: ) "(" (Set (Var "x")) "," (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" )))) ")" )))); theorem :: RSSPACE4:13 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "holds" (Bool (Set ($#k6_rsspace4 :::"modetrans"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r2_funct_2 :::"="::: ) (Set (Var "f")))))) ; theorem :: RSSPACE4:14 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "holds" (Bool (Set (Set "(" ($#k8_rsspace4 :::"BoundedFunctionsNorm"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k7_rsspace4 :::"PreNorms"::: ) (Set (Var "f")) ")" )))))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); func :::"R_NormSpace_of_BoundedFunctions"::: "(" "X" "," "Y" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_normsp_1 :::"NORMSTR"::: ) equals :: RSSPACE4:def 10 (Set ($#g1_normsp_1 :::"NORMSTR"::: ) "(#" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace4 :::"BoundedFunctionsNorm"::: ) "(" "X" "," "Y" ")" ")" ) "#)" ); end; :: deftheorem defines :::"R_NormSpace_of_BoundedFunctions"::: RSSPACE4:def 10 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) "holds" (Bool (Set ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g1_normsp_1 :::"NORMSTR"::: ) "(#" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set "(" ($#k4_rsspace4 :::"BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k5_lopban_1 :::"RealVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace4 :::"BoundedFunctionsNorm"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "#)" )))); theorem :: RSSPACE4:15 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) "holds" (Bool (Set (Set (Var "X")) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "Y")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ))))) ; theorem :: RSSPACE4:16 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "g")) "being" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "st" (Bool (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) ))))))) ; theorem :: RSSPACE4:17 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) ))))) ; theorem :: RSSPACE4:18 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) ))))) ; theorem :: RSSPACE4:19 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "f9")) "," (Set (Var "g9")) "," (Set (Var "h9")) "being" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "st" (Bool (Bool (Set (Var "f9")) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "g9")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "h9")) ($#r1_hidden :::"="::: ) (Set (Var "h")))) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "g")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "g9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))))) ; theorem :: RSSPACE4:20 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "f9")) "," (Set (Var "h9")) "being" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "st" (Bool (Bool (Set (Var "f9")) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "h9")) ($#r1_hidden :::"="::: ) (Set (Var "h")))) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set (Var "f9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))))) ; theorem :: RSSPACE4:21 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ))) ")" & "(" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )))) "implies" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set (Var "a")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) ))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "f")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "g")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) ) ($#k7_real_1 :::"+"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "g")) ($#k1_normsp_0 :::".||"::: ) ))) ")" ))))) ; theorem :: RSSPACE4:22 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) "holds" (Bool "(" (Bool (Set ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v4_normsp_0 :::"reflexive"::: ) ) & (Bool (Set ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v3_normsp_0 :::"discerning"::: ) ) & (Bool (Set ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v2_normsp_1 :::"RealNormSpace-like"::: ) ) ")" ))) ; theorem :: RSSPACE4:23 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) "holds" (Bool (Set ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#l1_normsp_1 :::"RealNormSpace":::)))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster (Set ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" "X" "," "Y" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#v3_normsp_0 :::"discerning"::: ) ($#v4_normsp_0 :::"reflexive"::: ) ($#v2_normsp_1 :::"RealNormSpace-like"::: ) ; end; theorem :: RSSPACE4:24 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "f9")) "," (Set (Var "g9")) "," (Set (Var "h9")) "being" ($#v1_rsspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "st" (Bool (Bool (Set (Var "f9")) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "g9")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "h9")) ($#r1_hidden :::"="::: ) (Set (Var "h")))) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "g")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "g9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))))) ; theorem :: RSSPACE4:25 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v3_lopban_1 :::"complete"::: ) )) "holds" (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set "(" ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_rsspace3 :::"Cauchy_sequence_by_Norm"::: ) )) "holds" (Bool (Set (Var "seq")) "is" ($#v3_normsp_1 :::"convergent"::: ) )))) ; theorem :: RSSPACE4:26 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_normsp_1 :::"RealBanachSpace":::) "holds" (Bool (Set ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#l1_normsp_1 :::"RealBanachSpace":::)))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_normsp_1 :::"RealBanachSpace":::); cluster (Set ($#k9_rsspace4 :::"R_NormSpace_of_BoundedFunctions"::: ) "(" "X" "," "Y" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lopban_1 :::"complete"::: ) ; end;