:: CSSPACE4 semantic presentation begin definitionfunc :::"the_set_of_BoundedComplexSequences"::: -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) means :: CSSPACE4: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_csspace :::"the_set_of_ComplexSequences"::: ) )) & (Bool (Set ($#k2_csspace :::"seq_id"::: ) (Set (Var "x"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) ")" )); end; :: deftheorem defines :::"the_set_of_BoundedComplexSequences"::: CSSPACE4:def 1 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) )) "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_csspace :::"the_set_of_ComplexSequences"::: ) )) & (Bool (Set ($#k2_csspace :::"seq_id"::: ) (Set (Var "x"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) ")" )) ")" )); registration cluster (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) -> ($#v6_clvect_1 :::"linearly-closed"::: ) ; end; registration cluster (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "#)" ) -> ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ; end; definitionfunc :::"Complex_linfty_norm"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: CSSPACE4:def 2 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set "(" ($#k2_csspace :::"seq_id"::: ) (Set (Var "x")) ")" ) ($#k55_valued_1 :::".|"::: ) ) ")" )))); end; :: deftheorem defines :::"Complex_linfty_norm"::: CSSPACE4:def 2 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k2_csspace4 :::"Complex_linfty_norm"::: ) )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ))) "holds" (Bool (Set (Set (Var "b1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set "(" ($#k2_csspace :::"seq_id"::: ) (Set (Var "x")) ")" ) ($#k55_valued_1 :::".|"::: ) ) ")" )))) ")" )); theorem :: CSSPACE4:1 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "seq")) ($#k55_valued_1 :::".|"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k5_complex1 :::"0c"::: ) ))) ")" )) ; registration cluster (Set ($#g2_clvect_1 :::"CNORMSTR"::: ) "(#" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set ($#k2_csspace4 :::"Complex_linfty_norm"::: ) ) "#)" ) -> ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ; end; definitionfunc :::"Complex_linfty_Space"::: -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_clvect_1 :::"CNORMSTR"::: ) equals :: CSSPACE4:def 3 (Set ($#g2_clvect_1 :::"CNORMSTR"::: ) "(#" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set ($#k2_csspace4 :::"Complex_linfty_norm"::: ) ) "#)" ); end; :: deftheorem defines :::"Complex_linfty_Space"::: CSSPACE4:def 3 : (Bool (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#g2_clvect_1 :::"CNORMSTR"::: ) "(#" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) ) "," (Set ($#k7_csspace :::"Linear_Space_of_ComplexSequences"::: ) ) ")" ")" ) "," (Set ($#k2_csspace4 :::"Complex_linfty_norm"::: ) ) "#)" )); theorem :: CSSPACE4:2 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_csspace4 :::"the_set_of_BoundedComplexSequences"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) )) "iff" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Complex_Sequence":::)) & (Bool (Set ($#k2_csspace :::"seq_id"::: ) (Set (Var "x"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) ")" ) ")" ) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_csspace :::"CZeroseq"::: ) )) & (Bool "(" "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) "holds" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k2_csspace :::"seq_id"::: ) (Set (Var "u")))) ")" ) & (Bool "(" "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) "holds" (Bool (Set (Set (Var "u")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_csspace :::"seq_id"::: ) (Set (Var "u")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" ($#k2_csspace :::"seq_id"::: ) (Set (Var "v")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) "holds" (Bool (Set (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" ($#k2_csspace :::"seq_id"::: ) (Set (Var "u")) ")" )))) ")" ) & (Bool "(" "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) "holds" (Bool "(" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k31_valued_1 :::"-"::: ) (Set "(" ($#k2_csspace :::"seq_id"::: ) (Set (Var "u")) ")" ))) & (Bool (Set ($#k2_csspace :::"seq_id"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "u")) ")" )) ($#r2_funct_2 :::"="::: ) (Set ($#k31_valued_1 :::"-"::: ) (Set "(" ($#k2_csspace :::"seq_id"::: ) (Set (Var "u")) ")" ))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) "holds" (Bool (Set (Set (Var "u")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_csspace :::"seq_id"::: ) (Set (Var "u")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" ($#k2_csspace :::"seq_id"::: ) (Set (Var "v")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) "holds" (Bool (Set ($#k2_csspace :::"seq_id"::: ) (Set (Var "v"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) & (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "v")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set "(" ($#k2_csspace :::"seq_id"::: ) (Set (Var "v")) ")" ) ($#k55_valued_1 :::".|"::: ) ) ")" ))) ")" ) ")" ) ; theorem :: CSSPACE4:3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) (Bool "for" (Set (Var "c")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool "(" "(" (Bool (Bool (Set ($#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_csspace4 :::"Complex_linfty_Space"::: ) ))) ")" & "(" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) )))) "implies" (Bool (Set ($#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 ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k17_complex1 :::"|."::: ) (Set (Var "c")) ($#k17_complex1 :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) ))) ")" ))) ; registration cluster (Set ($#k3_csspace4 :::"Complex_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"::: ) ($#v3_normsp_0 :::"discerning"::: ) ($#v4_normsp_0 :::"reflexive"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#v8_clvect_1 :::"ComplexNormSpace-like"::: ) ; end; theorem :: CSSPACE4:4 (Bool "for" (Set (Var "vseq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) "st" (Bool (Bool (Set (Var "vseq")) "is" ($#v1_csspace3 :::"Cauchy_sequence_by_Norm"::: ) )) "holds" (Bool (Set (Var "vseq")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) ; theorem :: CSSPACE4:5 (Bool (Set ($#k3_csspace4 :::"Complex_linfty_Space"::: ) ) "is" ($#l2_clvect_1 :::"ComplexBanachSpace":::)) ; begin definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "IT" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "Y"))); attr "IT" is :::"bounded"::: means :: CSSPACE4:def 4 (Bool "ex" (Set (Var "K")) "being" ($#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 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "K"))) ")" ) ")" )); end; :: deftheorem defines :::"bounded"::: CSSPACE4:def 4 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "IT")) "being" ($#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_csspace4 :::"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 :: CSSPACE4:6 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (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" ($#v1_csspace4 :::"bounded"::: ) )))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster ($#~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_csspace4 :::"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 ($#l2_clvect_1 :::"ComplexNormSpace":::); func :::"ComplexBoundedFunctions"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) means :: CSSPACE4:def 5 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) "is" ($#v1_csspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")) ")" )); end; :: deftheorem defines :::"ComplexBoundedFunctions"::: CSSPACE4:def 5 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (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" ($#v1_csspace4 :::"bounded"::: ) ($#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 ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster (Set ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: CSSPACE4:7 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) ))) ; theorem :: CSSPACE4:8 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "#)" ) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" )))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "#)" ) -> ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ; end; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); func :::"C_VectorSpace_of_BoundedFunctions"::: "(" "X" "," "Y" ")" -> ($#l1_clvect_1 :::"ComplexLinearSpace":::) equals :: CSSPACE4:def 6 (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "#)" ); end; :: deftheorem defines :::"C_VectorSpace_of_BoundedFunctions"::: CSSPACE4:def 6 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k5_csspace4 :::"C_VectorSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "#)" )))); registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster (Set ($#k5_csspace4 :::"C_VectorSpace_of_BoundedFunctions"::: ) "(" "X" "," "Y" ")" ) -> ($#v1_clvect_1 :::"strict"::: ) ; end; theorem :: CSSPACE4:9 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k5_csspace4 :::"C_VectorSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "f9")) "," (Set (Var "g9")) "," (Set (Var "h9")) "being" ($#v1_csspace4 :::"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")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))))) ; theorem :: CSSPACE4:10 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k5_csspace4 :::"C_VectorSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "f9")) "," (Set (Var "h9")) "being" ($#v1_csspace4 :::"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 "c")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#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 "c")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))))) ; theorem :: CSSPACE4:11 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k5_csspace4 :::"C_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 ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r2_hidden :::"in"::: ) (Set ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" )) ; func :::"modetrans"::: "(" "f" "," "X" "," "Y" ")" -> ($#v1_csspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") equals :: CSSPACE4:def 7 "f"; end; :: deftheorem defines :::"modetrans"::: CSSPACE4:def 7 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ))) "holds" (Bool (Set ($#k6_csspace4 :::"modetrans"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f")))))); definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); 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 :: CSSPACE4: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"::: CSSPACE4:def 8 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (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 ($#k7_csspace4 :::"PreNorms"::: ) (Set (Var "u"))) ($#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 :: CSSPACE4:12 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "g")) "being" ($#v1_csspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "holds" (Bool (Set ($#k7_csspace4 :::"PreNorms"::: ) (Set (Var "g"))) "is" ($#v4_xxreal_2 :::"bounded_above"::: ) )))) ; theorem :: CSSPACE4:13 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "g")) "being" ($#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_csspace4 :::"bounded"::: ) ) "iff" (Bool (Set ($#k7_csspace4 :::"PreNorms"::: ) (Set (Var "g"))) "is" ($#v4_xxreal_2 :::"bounded_above"::: ) ) ")" )))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); func :::"ComplexBoundedFunctionsNorm"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: CSSPACE4:def 9 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k7_csspace4 :::"PreNorms"::: ) (Set "(" ($#k6_csspace4 :::"modetrans"::: ) "(" (Set (Var "x")) "," "X" "," "Y" ")" ")" ) ")" )))); end; :: deftheorem defines :::"ComplexBoundedFunctionsNorm"::: CSSPACE4:def 9 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k8_csspace4 :::"ComplexBoundedFunctionsNorm"::: ) "(" (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 ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k7_csspace4 :::"PreNorms"::: ) (Set "(" ($#k6_csspace4 :::"modetrans"::: ) "(" (Set (Var "x")) "," (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" )))) ")" )))); theorem :: CSSPACE4:14 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#v1_csspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "holds" (Bool (Set ($#k6_csspace4 :::"modetrans"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r2_funct_2 :::"="::: ) (Set (Var "f")))))) ; theorem :: CSSPACE4:15 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#v1_csspace4 :::"bounded"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "holds" (Bool (Set (Set "(" ($#k8_csspace4 :::"ComplexBoundedFunctionsNorm"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k7_csspace4 :::"PreNorms"::: ) (Set (Var "f")) ")" )))))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); func :::"C_NormSpace_of_BoundedFunctions"::: "(" "X" "," "Y" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_clvect_1 :::"CNORMSTR"::: ) equals :: CSSPACE4:def 10 (Set ($#g2_clvect_1 :::"CNORMSTR"::: ) "(#" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace4 :::"ComplexBoundedFunctionsNorm"::: ) "(" "X" "," "Y" ")" ")" ) "#)" ); end; :: deftheorem defines :::"C_NormSpace_of_BoundedFunctions"::: CSSPACE4:def 10 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g2_clvect_1 :::"CNORMSTR"::: ) "(#" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k4_csspace4 :::"ComplexBoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace4 :::"ComplexBoundedFunctionsNorm"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "#)" )))); theorem :: CSSPACE4:16 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "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 "(" ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ))))) ; theorem :: CSSPACE4:17 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "g")) "being" ($#v1_csspace4 :::"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 :: CSSPACE4:18 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_csspace4 :::"C_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 :: CSSPACE4:19 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) ))))) ; theorem :: CSSPACE4:20 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "f9")) "," (Set (Var "g9")) "," (Set (Var "h9")) "being" ($#v1_csspace4 :::"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")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))))) ; theorem :: CSSPACE4:21 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "f9")) "," (Set (Var "h9")) "being" ($#v1_csspace4 :::"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 "c")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#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 "c")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f9")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))))) ; theorem :: CSSPACE4:22 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "c")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool "(" "(" (Bool (Bool (Set ($#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_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ))) ")" & "(" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )))) "implies" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k17_complex1 :::"|."::: ) (Set (Var "c")) ($#k17_complex1 :::".|"::: ) ) ($#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")) ($#k1_normsp_0 :::".||"::: ) ))) ")" ))))) ; theorem :: CSSPACE4:23 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool "(" (Bool (Set ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v4_normsp_0 :::"reflexive"::: ) ) & (Bool (Set ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v3_normsp_0 :::"discerning"::: ) ) & (Bool (Set ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v8_clvect_1 :::"ComplexNormSpace-like"::: ) ) ")" ))) ; theorem :: CSSPACE4:24 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#l2_clvect_1 :::"ComplexNormSpace":::)))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster (Set ($#k9_csspace4 :::"C_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"::: ) ($#v3_normsp_0 :::"discerning"::: ) ($#v4_normsp_0 :::"reflexive"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#v8_clvect_1 :::"ComplexNormSpace-like"::: ) ; end; theorem :: CSSPACE4:25 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "f9")) "," (Set (Var "g9")) "," (Set (Var "h9")) "being" ($#v1_csspace4 :::"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")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))))) ; theorem :: CSSPACE4:26 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v3_clopban1 :::"complete"::: ) )) "holds" (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set "(" ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_csspace3 :::"Cauchy_sequence_by_Norm"::: ) )) "holds" (Bool (Set (Var "seq")) "is" ($#v9_clvect_1 :::"convergent"::: ) )))) ; theorem :: CSSPACE4:27 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexBanachSpace":::) "holds" (Bool (Set ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#l2_clvect_1 :::"ComplexBanachSpace":::)))) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l2_clvect_1 :::"ComplexBanachSpace":::); cluster (Set ($#k9_csspace4 :::"C_NormSpace_of_BoundedFunctions"::: ) "(" "X" "," "Y" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_clopban1 :::"complete"::: ) ; end; begin theorem :: CSSPACE4:28 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "seq2")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set (Var "seq1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "seq2"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) ; theorem :: CSSPACE4:29 (Bool "for" (Set (Var "c")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set (Var "c")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "seq"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ))) ; theorem :: CSSPACE4:30 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) "iff" (Bool (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "seq")) ($#k55_valued_1 :::".|"::: ) ) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" )) ; theorem :: CSSPACE4:31 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "," (Set (Var "seq3")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "seq1")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "seq2")) ($#k46_valued_1 :::"-"::: ) (Set (Var "seq3")))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "seq2")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" (Set (Var "seq3")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" )))) ")" )) ;