:: CLOPBAN1 semantic presentation begin definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "Y" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "F" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set (Const "Y")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Const "Y")); let "c" be ($#v1_xcmplx_0 :::"complex"::: ) ($#m1_hidden :::"number"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "Y")); :: original: :::"[;]"::: redefine func "F" :::"[;]"::: "(" "c" "," "f" ")" -> ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," "Y" ")" ); end; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); func :::"FuncExtMult"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ")" ")" ) means :: CLOPBAN1:def 1 (Bool "for" (Set (Var "c")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "f")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "holds" (Bool (Set (Set "(" it ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "c")) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))))); end; :: deftheorem defines :::"FuncExtMult"::: CLOPBAN1:def 1 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" )) "iff" (Bool "for" (Set (Var "c")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "f")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "c")) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))))) ")" )))); theorem :: CLOPBAN1:1 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" ($#k4_lopban_1 :::"FuncZero"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "Y"))))))) ; theorem :: CLOPBAN1:2 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "h")) "," (Set (Var "f")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ) (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "a")) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) ))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))))) ; theorem :: CLOPBAN1:3 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ) "holds" (Bool (Set (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funcsdom :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ) ($#r2_funct_2 :::"="::: ) (Set (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funcsdom :::"."::: ) "(" (Set (Var "g")) "," (Set (Var "f")) ")" ))))) ; theorem :: CLOPBAN1:4 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ) "holds" (Bool (Set (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funcsdom :::"."::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funcsdom :::"."::: ) "(" (Set (Var "g")) "," (Set (Var "h")) ")" ")" ) ")" ) ($#r2_funct_2 :::"="::: ) (Set (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funcsdom :::"."::: ) "(" (Set "(" (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funcsdom :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ")" ) "," (Set (Var "h")) ")" ))))) ; theorem :: CLOPBAN1:5 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ) "holds" (Bool (Set (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funcsdom :::"."::: ) "(" (Set "(" ($#k4_lopban_1 :::"FuncZero"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set (Var "f")) ")" ) ($#r2_funct_2 :::"="::: ) (Set (Var "f")))))) ; theorem :: CLOPBAN1:6 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ) "holds" (Bool (Set (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funcsdom :::"."::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k2_funcsdom :::"."::: ) (Set ($#k1_domain_1 :::"["::: ) (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) "," (Set (Var "f")) ($#k1_domain_1 :::"]"::: ) ) ")" ) ")" ) ($#r2_funct_2 :::"="::: ) (Set ($#k4_lopban_1 :::"FuncZero"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ))))) ; theorem :: CLOPBAN1:7 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ) "holds" (Bool (Set (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k2_funcsdom :::"."::: ) (Set ($#k1_domain_1 :::"["::: ) (Set ($#k6_complex1 :::"1r"::: ) ) "," (Set (Var "f")) ($#k1_domain_1 :::"]"::: ) )) ($#r2_funct_2 :::"="::: ) (Set (Var "f")))))) ; theorem :: CLOPBAN1:8 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "a")) "," (Set "(" (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "b")) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) ) ")" ) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "b")) ")" ) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) ))))))) ; theorem :: CLOPBAN1:9 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "a")) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) ) ")" ) "," (Set "(" (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "b")) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "a")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "b")) ")" ) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) ))))))) ; theorem :: CLOPBAN1:10 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ")" ) "," (Set "(" ($#k4_lopban_1 :::"FuncZero"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "#)" ) "is" ($#l1_clvect_1 :::"ComplexLinearSpace":::)))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); func :::"ComplexVectSpace"::: "(" "X" "," "Y" ")" -> ($#l1_clvect_1 :::"ComplexLinearSpace":::) equals :: CLOPBAN1:def 2 (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ")" ")" ) "," (Set "(" ($#k4_lopban_1 :::"FuncZero"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" "X" "," "Y" ")" ")" ) "#)" ); end; :: deftheorem defines :::"ComplexVectSpace"::: CLOPBAN1:def 2 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ")" ) "," (Set "(" ($#k4_lopban_1 :::"FuncZero"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k2_lopban_1 :::"FuncAdd"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k2_clopban1 :::"FuncExtMult"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "#)" )))); registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); cluster (Set ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ) -> ($#v1_clvect_1 :::"strict"::: ) ; end; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); cluster (Set ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" "X" "," "Y" ")" ) -> ($#v1_monoid_0 :::"constituted-Functions"::: ) ; end; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); let "f" be ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" ")" ); let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); :: original: :::"."::: redefine func "f" :::"."::: "x" -> ($#m1_subset_1 :::"VECTOR":::) "of" "Y"; end; theorem :: CLOPBAN1:11 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "h")) ($#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 "h")) ($#k4_clopban1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k4_clopban1 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "g")) ($#k4_clopban1 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))) ; theorem :: CLOPBAN1:12 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "c")) "being" ($#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" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k4_clopban1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k4_clopban1 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))))) ; begin definitionlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "IT" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "Y")); attr "IT" is :::"homogeneous"::: means :: CLOPBAN1:def 3 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "X" (Bool "for" (Set (Var "r")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set "IT" ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "r")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_clvect_1 :::"*"::: ) (Set "(" "IT" ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ))))); end; :: deftheorem defines :::"homogeneous"::: CLOPBAN1:def 3 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "IT")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_clopban1 :::"homogeneous"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "IT")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "r")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "IT")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ))))) ")" ))); registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v13_vectsp_1 :::"additive"::: ) ($#v1_clopban1 :::"homogeneous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "X", "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); mode LinearOperator of "X" "," "Y" is ($#v13_vectsp_1 :::"additive"::: ) ($#v1_clopban1 :::"homogeneous"::: ) ($#m1_subset_1 :::"Function":::) "of" "X" "," "Y"; end; definitionlet "X", "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); func :::"LinearOperators"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," "Y" ")" ")" ) means :: CLOPBAN1:def 4 (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" ($#m1_subset_1 :::"LinearOperator":::) "of" "X" "," "Y") ")" )); end; :: deftheorem defines :::"LinearOperators"::: CLOPBAN1:def 4 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set (Var "Y")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k5_clopban1 :::"LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y"))) ")" )) ")" ))); registrationlet "X", "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); cluster (Set ($#k5_clopban1 :::"LinearOperators"::: ) "(" "X" "," "Y" ")" ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) ; end; theorem :: CLOPBAN1:13 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#k5_clopban1 :::"LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) )) ; theorem :: CLOPBAN1:14 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "#)" ) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set (Var "Y")) ")" ))) ; registrationlet "X", "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); cluster (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," "Y" ")" ")" ) ")" ")" ) "#)" ) -> ($#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"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ; end; definitionlet "X", "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); func :::"C_VectorSpace_of_LinearOperators"::: "(" "X" "," "Y" ")" -> ($#l1_clvect_1 :::"ComplexLinearSpace":::) equals :: CLOPBAN1:def 5 (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," "Y" ")" ")" ) ")" ")" ) "#)" ); end; :: deftheorem defines :::"C_VectorSpace_of_LinearOperators"::: CLOPBAN1:def 5 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k5_clopban1 :::"LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k3_clopban1 :::"ComplexVectSpace"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "#)" ))); registrationlet "X", "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); cluster (Set ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ) -> ($#v1_clvect_1 :::"strict"::: ) ; end; registrationlet "X", "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); cluster (Set ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ) -> ($#v1_monoid_0 :::"constituted-Functions"::: ) ; end; definitionlet "X", "Y" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); let "f" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" ")" ); let "v" be ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Const "X")); :: original: :::"."::: redefine func "f" :::"."::: "v" -> ($#m1_subset_1 :::"VECTOR":::) "of" "Y"; end; theorem :: CLOPBAN1:15 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "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" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k7_clopban1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_clopban1 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "g")) ($#k7_clopban1 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))) ; theorem :: CLOPBAN1:16 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "c")) "being" ($#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" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k7_clopban1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k7_clopban1 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))) ; theorem :: CLOPBAN1:17 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "Y")) ")" )))) ; theorem :: CLOPBAN1:18 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "Y")) ")" )) "is" ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y")))) ; begin theorem :: CLOPBAN1:19 (Bool "for" (Set (Var "X")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v9_clvect_1 :::"convergent"::: ) ) & (Bool (Set ($#k7_clvect_1 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set ($#k4_normsp_0 :::"||."::: ) (Set (Var "seq")) ($#k4_normsp_0 :::".||"::: ) ) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set ($#k4_normsp_0 :::"||."::: ) (Set (Var "seq")) ($#k4_normsp_0 :::".||"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "g")) ($#k1_normsp_0 :::".||"::: ) )) ")" )))) ; definitionlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "IT" be ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Const "X")) "," (Set (Const "Y")); attr "IT" is :::"Lipschitzian"::: means :: CLOPBAN1:def 6 (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 :::"VECTOR":::) "of" "X" "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" "IT" ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "K")) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) ))) ")" ) ")" )); end; :: deftheorem defines :::"Lipschitzian"::: CLOPBAN1:def 6 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "IT")) "being" ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v2_clopban1 :::"Lipschitzian"::: ) ) "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 :::"VECTOR":::) "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 (Set (Var "K")) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) ))) ")" ) ")" )) ")" ))); theorem :: CLOPBAN1:20 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "Y")))) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v2_clopban1 :::"Lipschitzian"::: ) ))) ; registrationlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v13_vectsp_1 :::"additive"::: ) ($#v1_clopban1 :::"homogeneous"::: ) ($#v2_clopban1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); func :::"BoundedLinearOperators"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) means :: CLOPBAN1:def 7 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) "is" ($#v2_clopban1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"LinearOperator":::) "of" "X" "," "Y") ")" )); end; :: deftheorem defines :::"BoundedLinearOperators"::: CLOPBAN1:def 7 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (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" ($#v2_clopban1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y"))) ")" )) ")" ))); registrationlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster (Set ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: CLOPBAN1:21 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) )) ; theorem :: CLOPBAN1:22 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "#)" ) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ))) ; registrationlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "#)" ) -> ($#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"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ; end; definitionlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); func :::"C_VectorSpace_of_BoundedLinearOperators"::: "(" "X" "," "Y" ")" -> ($#l1_clvect_1 :::"ComplexLinearSpace":::) equals :: CLOPBAN1:def 8 (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "#)" ); end; :: deftheorem defines :::"C_VectorSpace_of_BoundedLinearOperators"::: CLOPBAN1:def 8 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k9_clopban1 :::"C_VectorSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "#)" ))); registrationlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster (Set ($#k9_clopban1 :::"C_VectorSpace_of_BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ) -> ($#v1_clvect_1 :::"strict"::: ) ; end; registrationlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k9_clopban1 :::"C_VectorSpace_of_BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" )); end; definitionlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k9_clopban1 :::"C_VectorSpace_of_BoundedLinearOperators"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" ")" ); let "v" be ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Const "X")); :: original: :::"."::: redefine func "f" :::"."::: "v" -> ($#m1_subset_1 :::"VECTOR":::) "of" "Y"; end; theorem :: CLOPBAN1:23 (Bool "for" (Set (Var "X")) "," (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 "(" ($#k9_clopban1 :::"C_VectorSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "g")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k10_clopban1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k10_clopban1 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "g")) ($#k10_clopban1 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))) ; theorem :: CLOPBAN1:24 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k9_clopban1 :::"C_VectorSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (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 :::"VECTOR":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k10_clopban1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k10_clopban1 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))) ; theorem :: CLOPBAN1:25 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k9_clopban1 :::"C_VectorSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "Y")) ")" )))) ; definitionlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" )) ; func :::"modetrans"::: "(" "f" "," "X" "," "Y" ")" -> ($#v2_clopban1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"LinearOperator":::) "of" "X" "," "Y" equals :: CLOPBAN1:def 9 "f"; end; :: deftheorem defines :::"modetrans"::: CLOPBAN1:def 9 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ))) "holds" (Bool (Set ($#k11_clopban1 :::"modetrans"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f"))))); definitionlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "u" be ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Const "X")) "," (Set (Const "Y")); func :::"PreNorms"::: "u" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: CLOPBAN1:def 10 "{" (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" "u" ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) where t "is" ($#m1_subset_1 :::"VECTOR":::) "of" "X" : (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "t")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) "}" ; end; :: deftheorem defines :::"PreNorms"::: CLOPBAN1:def 10 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool (Set ($#k12_clopban1 :::"PreNorms"::: ) (Set (Var "u"))) ($#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 :::"VECTOR":::) "of" (Set (Var "X")) : (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "t")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) "}" ))); theorem :: CLOPBAN1:26 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "g")) "being" ($#v2_clopban1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool (Set ($#k12_clopban1 :::"PreNorms"::: ) (Set (Var "g"))) "is" ($#v4_xxreal_2 :::"bounded_above"::: ) ))) ; theorem :: CLOPBAN1:27 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v2_clopban1 :::"Lipschitzian"::: ) ) "iff" (Bool (Set ($#k12_clopban1 :::"PreNorms"::: ) (Set (Var "g"))) "is" ($#v4_xxreal_2 :::"bounded_above"::: ) ) ")" ))) ; definitionlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); func :::"BoundedLinearOperatorsNorm"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: CLOPBAN1:def 11 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k12_clopban1 :::"PreNorms"::: ) (Set "(" ($#k11_clopban1 :::"modetrans"::: ) "(" (Set (Var "x")) "," "X" "," "Y" ")" ")" ) ")" )))); end; :: deftheorem defines :::"BoundedLinearOperatorsNorm"::: CLOPBAN1:def 11 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k13_clopban1 :::"BoundedLinearOperatorsNorm"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k12_clopban1 :::"PreNorms"::: ) (Set "(" ($#k11_clopban1 :::"modetrans"::: ) "(" (Set (Var "x")) "," (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" )))) ")" ))); theorem :: CLOPBAN1:28 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#v2_clopban1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool (Set ($#k11_clopban1 :::"modetrans"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r2_funct_2 :::"="::: ) (Set (Var "f"))))) ; theorem :: CLOPBAN1:29 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#v2_clopban1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool (Set (Set "(" ($#k13_clopban1 :::"BoundedLinearOperatorsNorm"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k12_clopban1 :::"PreNorms"::: ) (Set (Var "f")) ")" ))))) ; definitionlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); func :::"C_NormSpace_of_BoundedLinearOperators"::: "(" "X" "," "Y" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_clvect_1 :::"CNORMSTR"::: ) equals :: CLOPBAN1:def 12 (Set ($#g2_clvect_1 :::"CNORMSTR"::: ) "(#" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" "X" "," "Y" ")" ")" ) ")" ")" ) "," (Set "(" ($#k13_clopban1 :::"BoundedLinearOperatorsNorm"::: ) "(" "X" "," "Y" ")" ")" ) "#)" ); end; :: deftheorem defines :::"C_NormSpace_of_BoundedLinearOperators"::: CLOPBAN1:def 12 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g2_clvect_1 :::"CNORMSTR"::: ) "(#" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_csspace :::"Zero_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k8_csspace :::"Add_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k9_csspace :::"Mult_"::: ) "(" (Set "(" ($#k8_clopban1 :::"BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k6_clopban1 :::"C_VectorSpace_of_LinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ")" ")" ) "," (Set "(" ($#k13_clopban1 :::"BoundedLinearOperatorsNorm"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "#)" ))); theorem :: CLOPBAN1:30 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "Y")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )))) ; theorem :: CLOPBAN1:31 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "g")) "being" ($#v2_clopban1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"LinearOperator":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "st" (Bool (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "t")) ($#k1_normsp_0 :::".||"::: ) ))))))) ; theorem :: CLOPBAN1:32 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) )))) ; theorem :: CLOPBAN1:33 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k1_normsp_0 :::".||"::: ) )))) ; registrationlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ")" )); end; definitionlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Const "X")) "," (Set (Const "Y")) ")" ")" ); let "v" be ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Const "X")); :: original: :::"."::: redefine func "f" :::"."::: "v" -> ($#m1_subset_1 :::"VECTOR":::) "of" "Y"; end; theorem :: CLOPBAN1:34 (Bool "for" (Set (Var "X")) "," (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 "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "g")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k15_clopban1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k15_clopban1 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "g")) ($#k15_clopban1 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))) ; theorem :: CLOPBAN1:35 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (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 :::"VECTOR":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k15_clopban1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k15_clopban1 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))) ; theorem :: CLOPBAN1:36 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) (Bool "for" (Set (Var "c")) "being" ($#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 "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ))) ")" & "(" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (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 :: CLOPBAN1:37 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool "(" (Bool (Set ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v4_normsp_0 :::"reflexive"::: ) ) & (Bool (Set ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v3_normsp_0 :::"discerning"::: ) ) & (Bool (Set ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#v8_clvect_1 :::"ComplexNormSpace-like"::: ) ) ")" )) ; theorem :: CLOPBAN1:38 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#l2_clvect_1 :::"ComplexNormSpace":::))) ; registrationlet "X", "Y" be ($#l2_clvect_1 :::"ComplexNormSpace":::); cluster (Set ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" "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"::: ) ($#v8_clvect_1 :::"ComplexNormSpace-like"::: ) ; end; theorem :: CLOPBAN1:39 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "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" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k15_clopban1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k15_clopban1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "g")) ($#k15_clopban1 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))) ; begin definitionlet "X" be ($#l2_clvect_1 :::"ComplexNormSpace":::); attr "X" is :::"complete"::: means :: CLOPBAN1:def 13 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" "X" "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_csspace3 :::"Cauchy_sequence_by_Norm"::: ) )) "holds" (Bool (Set (Var "seq")) "is" ($#v9_clvect_1 :::"convergent"::: ) )); end; :: deftheorem defines :::"complete"::: CLOPBAN1:def 13 : (Bool "for" (Set (Var "X")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v3_clopban1 :::"complete"::: ) ) "iff" (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_csspace3 :::"Cauchy_sequence_by_Norm"::: ) )) "holds" (Bool (Set (Var "seq")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) ")" )); registration cluster ($#~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"::: ) ($#v3_clopban1 :::"complete"::: ) for ($#l2_clvect_1 :::"CNORMSTR"::: ) ; end; definitionmode ComplexBanachSpace is ($#v3_clopban1 :::"complete"::: ) ($#l2_clvect_1 :::"ComplexNormSpace":::); end; theorem :: CLOPBAN1:40 (Bool "for" (Set (Var "X")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool "(" (Bool (Set ($#k4_normsp_0 :::"||."::: ) (Set (Var "seq")) ($#k4_normsp_0 :::".||"::: ) ) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set ($#k4_normsp_0 :::"||."::: ) (Set (Var "seq")) ($#k4_normsp_0 :::".||"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" ($#k7_clvect_1 :::"lim"::: ) (Set (Var "seq")) ")" ) ($#k1_normsp_0 :::".||"::: ) )) ")" ))) ; theorem :: CLOPBAN1:41 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#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 "(" ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_csspace3 :::"Cauchy_sequence_by_Norm"::: ) )) "holds" (Bool (Set (Var "seq")) "is" ($#v9_clvect_1 :::"convergent"::: ) ))) ; theorem :: CLOPBAN1:42 (Bool "for" (Set (Var "X")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#l2_clvect_1 :::"ComplexBanachSpace":::) "holds" (Bool (Set ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) "is" ($#l2_clvect_1 :::"ComplexBanachSpace":::)))) ; registrationlet "X" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "Y" be ($#l2_clvect_1 :::"ComplexBanachSpace":::); cluster (Set ($#k14_clopban1 :::"C_NormSpace_of_BoundedLinearOperators"::: ) "(" "X" "," "Y" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_clopban1 :::"complete"::: ) ; end;