:: CLVECT_1 semantic presentation begin definitionattr "c1" is :::"strict"::: ; struct :::"CLSStruct"::: -> ($#l2_algstr_0 :::"addLoopStr"::: ) ; aggr :::"CLSStruct":::(# :::"carrier":::, :::"ZeroF":::, :::"addF":::, :::"Mult"::: #) -> ($#l1_clvect_1 :::"CLSStruct"::: ) ; sel :::"Mult"::: "c1" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1") ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1"); end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) for ($#l1_clvect_1 :::"CLSStruct"::: ) ; end; definitionlet "V" be ($#l1_clvect_1 :::"CLSStruct"::: ) ; mode VECTOR of "V" is ($#m1_subset_1 :::"Element":::) "of" "V"; end; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "v" be ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Const "V")); let "z" be ($#m1_hidden :::"Complex":::); func "z" :::"*"::: "v" -> ($#m1_subset_1 :::"Element":::) "of" "V" equals :: CLVECT_1:def 1 (Set (Set "the" ($#u1_clvect_1 :::"Mult"::: ) "of" "V") ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) "z" "," "v" ($#k4_tarski :::"]"::: ) )); end; :: deftheorem defines :::"*"::: CLVECT_1:def 1 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_clvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "z")) "," (Set (Var "v")) ($#k4_tarski :::"]"::: ) )))))); registrationlet "ZS" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "O" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "ZS")); let "F" be ($#m1_subset_1 :::"BinOp":::) "of" (Set (Const "ZS")); let "G" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set (Const "ZS")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Const "ZS")); cluster (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" "ZS" "," "O" "," "F" "," "G" "#)" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; definitionlet "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; attr "IT" is :::"vector-distributive"::: means :: CLVECT_1:def 2 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "IT" "holds" (Bool (Set (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "w")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "w")) ")" ))))); attr "IT" is :::"scalar-distributive"::: means :: CLVECT_1:def 3 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "IT" "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "b")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ))))); attr "IT" is :::"scalar-associative"::: means :: CLVECT_1:def 4 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "IT" "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "b")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ))))); attr "IT" is :::"scalar-unital"::: means :: CLVECT_1:def 5 (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "IT" "holds" (Bool (Set (Set ($#k6_complex1 :::"1r"::: ) ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Var "v")))); end; :: deftheorem defines :::"vector-distributive"::: CLVECT_1:def 2 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v2_clvect_1 :::"vector-distributive"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "w")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "w")) ")" ))))) ")" )); :: deftheorem defines :::"scalar-distributive"::: CLVECT_1:def 3 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v3_clvect_1 :::"scalar-distributive"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "b")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ))))) ")" )); :: deftheorem defines :::"scalar-associative"::: CLVECT_1:def 4 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v4_clvect_1 :::"scalar-associative"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "b")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ))))) ")" )); :: deftheorem defines :::"scalar-unital"::: CLVECT_1:def 5 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v5_clvect_1 :::"scalar-unital"::: ) ) "iff" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "IT")) "holds" (Bool (Set (Set ($#k6_complex1 :::"1r"::: ) ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Var "v")))) ")" )); definitionfunc :::"Trivial-CLSStruct"::: -> ($#v1_clvect_1 :::"strict"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) equals :: CLVECT_1:def 6 (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Num 1) "," (Set ($#k5_funct_5 :::"op0"::: ) ) "," (Set ($#k9_funct_5 :::"op2"::: ) ) "," (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Num 1) ")" ")" ) "#)" ); end; :: deftheorem defines :::"Trivial-CLSStruct"::: CLVECT_1:def 6 : (Bool (Set ($#k2_clvect_1 :::"Trivial-CLSStruct"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Num 1) "," (Set ($#k5_funct_5 :::"op0"::: ) ) "," (Set ($#k9_funct_5 :::"op2"::: ) ) "," (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Num 1) ")" ")" ) "#)" )); registration cluster (Set ($#k2_clvect_1 :::"Trivial-CLSStruct"::: ) ) -> (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v1_clvect_1 :::"strict"::: ) ; end; 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"::: ) ($#v1_clvect_1 :::"strict"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) for ($#l1_clvect_1 :::"CLSStruct"::: ) ; end; definitionmode ComplexLinearSpace is ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; end; theorem :: CLVECT_1:1 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) ")" )) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))))))) ; theorem :: CLVECT_1:2 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool "(" "not" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) "or" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) ")" )))) ; theorem :: CLVECT_1:3 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")))))) ; theorem :: CLVECT_1:4 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))))) "holds" (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))))) ; theorem :: CLVECT_1:5 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))))) "holds" (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))))) ; theorem :: CLVECT_1:6 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "z")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))))))) ; theorem :: CLVECT_1:7 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" )))))) ; theorem :: CLVECT_1:8 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "z")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))))))) ; theorem :: CLVECT_1:9 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "u")) ")" )))))) ; theorem :: CLVECT_1:10 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z1")) "," (Set (Var "z2")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set "(" (Set (Var "z1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "z2")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z1")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "z2")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" )))))) ; theorem :: CLVECT_1:11 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "u"))))) "holds" (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set (Var "u")))))) ; theorem :: CLVECT_1:12 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z1")) "," (Set (Var "z2")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool (Set (Var "v")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) & (Bool (Set (Set (Var "z1")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "z2")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))))) "holds" (Bool (Set (Var "z1")) ($#r1_hidden :::"="::: ) (Set (Var "z2")))))) ; theorem :: CLVECT_1:13 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "F")))) & (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))))) ")" )) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "G")) ")" )))))) ; theorem :: CLVECT_1:14 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k6_finseq_1 :::"<*>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))))) ; theorem :: CLVECT_1:15 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "v")) "," (Set (Var "u")) ($#k2_finseq_4 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "u")) ")" )))))) ; theorem :: CLVECT_1:16 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "u")) "," (Set (Var "v1")) "," (Set (Var "v2")) ($#k3_finseq_4 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "u")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v2")) ")" )))))) ; theorem :: CLVECT_1:17 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "v")) "," (Set (Var "v")) ($#k2_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k8_complex1 :::"+"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")))))) ; theorem :: CLVECT_1:18 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) "," (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k2_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k8_complex1 :::"+"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")))))) ; theorem :: CLVECT_1:19 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "v")) "," (Set (Var "v")) "," (Set (Var "v")) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k8_complex1 :::"+"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k8_complex1 :::"+"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")))))) ; begin definitionlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); let "V1" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); attr "V1" is :::"linearly-closed"::: means :: CLVECT_1:def 7 (Bool "(" (Bool "(" "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) "V1") & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "V1")) "holds" (Bool (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) "V1") ")" ) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) "V1")) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) "V1")) ")" ) ")" ); end; :: deftheorem defines :::"linearly-closed"::: CLVECT_1:def 7 : (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "V1")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) ) "iff" (Bool "(" (Bool "(" "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) "holds" (Bool (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))) ")" ) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) ")" ) ")" ) ")" ))); theorem :: CLVECT_1:20 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "V1")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) )) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))))) ; theorem :: CLVECT_1:21 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) )) "holds" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))))) ; theorem :: CLVECT_1:22 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) )) "holds" (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) "holds" (Bool (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))))) ; theorem :: CLVECT_1:23 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) )) ; theorem :: CLVECT_1:24 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set (Var "V1")))) "holds" (Bool (Set (Var "V1")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) ))) ; theorem :: CLVECT_1:25 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "," (Set (Var "V2")) "," (Set (Var "V3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) ) & (Bool (Set (Var "V2")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) ) & (Bool (Set (Var "V3")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")) ")" ) where v, u "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) ")" ) "}" )) "holds" (Bool (Set (Var "V3")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) ))) ; theorem :: CLVECT_1:26 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "," (Set (Var "V2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) ) & (Bool (Set (Var "V2")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) )) "holds" (Bool (Set (Set (Var "V1")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "V2"))) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) ))) ; definitionlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); mode :::"Subspace"::: "of" "V" -> ($#l1_clvect_1 :::"ComplexLinearSpace":::) means :: CLVECT_1:def 8 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V")) & (Bool (Set ($#k4_struct_0 :::"0."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "V")) & (Bool (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" "V") ($#k1_realset1 :::"||"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it))) & (Bool (Set "the" ($#u1_clvect_1 :::"Mult"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_clvect_1 :::"Mult"::: ) "of" "V") ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#k2_zfmisc_1 :::":]"::: ) ))) ")" ); end; :: deftheorem defines :::"Subspace"::: CLVECT_1:def 8 : (Bool "for" (Set (Var "V")) "," (Set (Var "b2")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V"))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) & (Bool (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "V"))) ($#k1_realset1 :::"||"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))))) & (Bool (Set "the" ($#u1_clvect_1 :::"Mult"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_clvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#k2_zfmisc_1 :::":]"::: ) ))) ")" ) ")" )); theorem :: CLVECT_1:27 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W1"))) & (Bool (Set (Var "W1")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "W2")))) "holds" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W2")))))) ; theorem :: CLVECT_1:28 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "V")))))) ; theorem :: CLVECT_1:29 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "W")) "holds" (Bool (Set (Var "w")) "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")))))) ; theorem :: CLVECT_1:30 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))))) ; theorem :: CLVECT_1:31 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W2")))))) ; theorem :: CLVECT_1:32 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "w1")) ($#r1_hidden :::"="::: ) (Set (Var "v"))) & (Bool (Set (Var "w2")) ($#r1_hidden :::"="::: ) (Set (Var "u")))) "holds" (Bool (Set (Set (Var "w1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "w2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")))))))) ; theorem :: CLVECT_1:33 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "w")) ($#r1_hidden :::"="::: ) (Set (Var "v")))) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))))))))) ; theorem :: CLVECT_1:34 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "w")) ($#r1_hidden :::"="::: ) (Set (Var "v")))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "w")))))))) ; theorem :: CLVECT_1:35 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "w1")) ($#r1_hidden :::"="::: ) (Set (Var "v"))) & (Bool (Set (Var "w2")) ($#r1_hidden :::"="::: ) (Set (Var "u")))) "holds" (Bool (Set (Set (Var "w1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u")))))))) ; theorem :: CLVECT_1:36 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))))) ; theorem :: CLVECT_1:37 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W1"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W2"))))) ; theorem :: CLVECT_1:38 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "V"))))) ; theorem :: CLVECT_1:39 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set (Var "u")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))))) ; theorem :: CLVECT_1:40 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))))))) ; theorem :: CLVECT_1:41 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))))) ; theorem :: CLVECT_1:42 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set (Var "u")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))))) ; theorem :: CLVECT_1:43 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set (Var "D")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "D")) "st" (Bool (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool (Set (Var "d1")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "V"))) ($#k1_realset1 :::"||"::: ) (Set (Var "V1")))) & (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_clvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set (Var "V1")) ($#k2_zfmisc_1 :::":]"::: ) )))) "holds" (Bool (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set (Var "D")) "," (Set (Var "d1")) "," (Set (Var "A")) "," (Set (Var "M")) "#)" ) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V"))))))))) ; theorem :: CLVECT_1:44 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set (Var "V")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")))) ; theorem :: CLVECT_1:45 (Bool "for" (Set (Var "V")) "," (Set (Var "X")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#l1_clvect_1 :::"ComplexLinearSpace":::) "st" (Bool (Bool (Set (Var "V")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "X"))) & (Bool (Set (Var "X")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")))) "holds" (Bool (Set (Var "V")) ($#r1_hidden :::"="::: ) (Set (Var "X")))) ; theorem :: CLVECT_1:46 (Bool "for" (Set (Var "V")) "," (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "st" (Bool (Bool (Set (Var "V")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "X"))) & (Bool (Set (Var "X")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "Y")))) "holds" (Bool (Set (Var "V")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "Y")))) ; theorem :: CLVECT_1:47 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W1"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W2"))))) "holds" (Bool (Set (Var "W1")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "W2"))))) ; theorem :: CLVECT_1:48 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W1")))) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W2"))) ")" )) "holds" (Bool (Set (Var "W1")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "W2"))))) ; registrationlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); 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"::: ) ($#v1_clvect_1 :::"strict"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) for ($#m1_clvect_1 :::"Subspace"::: ) "of" "V"; end; theorem :: CLVECT_1:49 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W2"))))) "holds" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))))) ; theorem :: CLVECT_1:50 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W1"))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W2"))) ")" ) ")" )) "holds" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))))) ; theorem :: CLVECT_1:51 (Bool "for" (Set (Var "V")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))))) "holds" (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "V"))))) ; theorem :: CLVECT_1:52 (Bool "for" (Set (Var "V")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "V"))) ")" ) ")" )) "holds" (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "V"))))) ; theorem :: CLVECT_1:53 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Var "V1")))) "holds" (Bool (Set (Var "V1")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) )))) ; theorem :: CLVECT_1:54 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "V1")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) )) "holds" (Bool "ex" (Set (Var "W")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))))) ; definitionlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); func :::"(0)."::: "V" -> ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" "V" means :: CLVECT_1:def 9 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "V" ")" ) ($#k6_domain_1 :::"}"::: ) )); end; :: deftheorem defines :::"(0)."::: CLVECT_1:def 9 : (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "b2")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "V")))) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ")" ))); definitionlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); func :::"(Omega)."::: "V" -> ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" "V" equals :: CLVECT_1:def 10 (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" "V") "," (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" "V") "," (Set "the" ($#u1_clvect_1 :::"Mult"::: ) "of" "V") "#)" ); end; :: deftheorem defines :::"(Omega)."::: CLVECT_1:def 10 : (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#k4_clvect_1 :::"(Omega)."::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_clvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) "#)" ))); theorem :: CLVECT_1:55 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "V")))))) ; theorem :: CLVECT_1:56 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "W2")))))) ; theorem :: CLVECT_1:57 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "W"))) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V"))))) ; theorem :: CLVECT_1:58 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "V"))) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "W"))))) ; theorem :: CLVECT_1:59 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "W1"))) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "W2"))))) ; theorem :: CLVECT_1:60 (Bool "for" (Set (Var "V")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set (Var "V")) "is" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set ($#k4_clvect_1 :::"(Omega)."::: ) (Set (Var "V"))))) ; definitionlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); let "v" be ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Const "V")); let "W" be ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Const "V")); func "v" :::"+"::: "W" -> ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: CLVECT_1:def 11 "{" (Set "(" "v" ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")) ")" ) where u "is" ($#m1_subset_1 :::"VECTOR":::) "of" "V" : (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) "W") "}" ; end; :: deftheorem defines :::"+"::: CLVECT_1:def 11 : (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")) ")" ) where u "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "}" )))); definitionlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); let "W" be ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Const "V")); mode :::"Coset"::: "of" "W" -> ($#m1_subset_1 :::"Subset":::) "of" "V" means :: CLVECT_1:def 12 (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" "st" (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) "W"))); end; :: deftheorem defines :::"Coset"::: CLVECT_1:def 12 : (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W"))) "iff" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))))) ")" )))); theorem :: CLVECT_1:61 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: CLVECT_1:62 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))))))) ; theorem :: CLVECT_1:63 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))))) ; theorem :: CLVECT_1:64 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set "(" ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: CLVECT_1:65 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set "(" ($#k4_clvect_1 :::"(Omega)."::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))))) ; theorem :: CLVECT_1:66 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" )))) ; theorem :: CLVECT_1:67 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" )))) ; theorem :: CLVECT_1:68 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))))))) ; theorem :: CLVECT_1:69 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))))))) ; theorem :: CLVECT_1:70 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" )))) ; theorem :: CLVECT_1:71 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")) ")" ) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) ")" )))) ; theorem :: CLVECT_1:72 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u")) ")" ) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) ")" )))) ; theorem :: CLVECT_1:73 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "u")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Set (Var "u")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) ")" )))) ; theorem :: CLVECT_1:74 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: CLVECT_1:75 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v1")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v2")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool (Set (Set (Var "v1")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v2")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))))))) ; theorem :: CLVECT_1:76 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))))) ; theorem :: CLVECT_1:77 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_complex1 :::"1r"::: ) )) & (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))))))) ; theorem :: CLVECT_1:78 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))))))) ; theorem :: CLVECT_1:79 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: CLVECT_1:80 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "u")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: CLVECT_1:81 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: CLVECT_1:82 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v1")))) ")" )) ")" )))) ; theorem :: CLVECT_1:83 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v1")))) ")" )) ")" )))) ; theorem :: CLVECT_1:84 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) & (Bool (Set (Var "v2")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) ")" )) "iff" (Bool (Set (Set (Var "v1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v2"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: CLVECT_1:85 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set (Var "u"))) ")" ))))) ; theorem :: CLVECT_1:86 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set (Var "u"))) ")" ))))) ; theorem :: CLVECT_1:87 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W2")))) "iff" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))) ")" )))) ; theorem :: CLVECT_1:88 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W2"))))) "holds" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2")))))) ; theorem :: CLVECT_1:89 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) ) "iff" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" )))) ; theorem :: CLVECT_1:90 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C1")) "being" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W1")) (Bool "for" (Set (Var "C2")) "being" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W2")) "st" (Bool (Bool (Set (Var "C1")) ($#r1_hidden :::"="::: ) (Set (Var "C2")))) "holds" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))))))) ; theorem :: CLVECT_1:91 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "V")))))) ; theorem :: CLVECT_1:92 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "V"))))) "holds" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))))) ; theorem :: CLVECT_1:93 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))) "is" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W"))))) ; theorem :: CLVECT_1:94 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "is" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set ($#k4_clvect_1 :::"(Omega)."::: ) (Set (Var "V"))))) ; theorem :: CLVECT_1:95 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set ($#k4_clvect_1 :::"(Omega)."::: ) (Set (Var "V"))))) "holds" (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))))) ; theorem :: CLVECT_1:96 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W")) "holds" (Bool "(" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "iff" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" )))) ; theorem :: CLVECT_1:97 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "iff" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k5_clvect_1 :::"+"::: ) (Set (Var "W")))) ")" ))))) ; theorem :: CLVECT_1:98 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "u")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set (Var "v"))) ")" )))))) ; theorem :: CLVECT_1:99 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "u")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set (Var "v"))) ")" )))))) ; theorem :: CLVECT_1:100 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool "ex" (Set (Var "C")) "being" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "v2")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )) "iff" (Bool (Set (Set (Var "v1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v2"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: CLVECT_1:101 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "B")) "," (Set (Var "C")) "being" ($#m2_clvect_1 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "B"))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set (Var "C"))))))) ; begin definitionattr "c1" is :::"strict"::: ; struct :::"CNORMSTR"::: -> ($#l1_clvect_1 :::"CLSStruct"::: ) "," ($#l2_normsp_0 :::"N-ZeroStr"::: ) ; aggr :::"CNORMSTR":::(# :::"carrier":::, :::"ZeroF":::, :::"addF":::, :::"Mult":::, :::"normF"::: #) -> ($#l2_clvect_1 :::"CNORMSTR"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) for ($#l2_clvect_1 :::"CNORMSTR"::: ) ; end; definitionlet "IT" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_clvect_1 :::"CNORMSTR"::: ) ; attr "IT" is :::"ComplexNormSpace-like"::: means :: CLVECT_1:def 13 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" "IT" (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool "(" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k17_complex1 :::"|."::: ) (Set (Var "z")) ($#k17_complex1 :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (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")) ($#k1_normsp_0 :::".||"::: ) ))) ")" ))); end; :: deftheorem defines :::"ComplexNormSpace-like"::: CLVECT_1:def 13 : (Bool "for" (Set (Var "IT")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_clvect_1 :::"CNORMSTR"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v8_clvect_1 :::"ComplexNormSpace-like"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "IT")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool "(" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k17_complex1 :::"|."::: ) (Set (Var "z")) ($#k17_complex1 :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (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")) ($#k1_normsp_0 :::".||"::: ) ))) ")" ))) ")" )); registration cluster ($#~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"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#v7_clvect_1 :::"strict"::: ) ($#v8_clvect_1 :::"ComplexNormSpace-like"::: ) for ($#l2_clvect_1 :::"CNORMSTR"::: ) ; end; definitionmode ComplexNormSpace is ($#~v2_struct_0 "non" ($#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"::: ) ($#v8_clvect_1 :::"ComplexNormSpace-like"::: ) ($#l2_clvect_1 :::"CNORMSTR"::: ) ; end; theorem :: CLVECT_1:102 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "CNS")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: CLVECT_1:103 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) )))) ; theorem :: CLVECT_1:104 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_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")) ($#k1_normsp_0 :::".||"::: ) ))))) ; theorem :: CLVECT_1:105 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) )))) ; theorem :: CLVECT_1:106 (Bool "for" (Set (Var "z1")) "," (Set (Var "z2")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" (Set (Var "z1")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "z2")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "y")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set ($#k17_complex1 :::"|."::: ) (Set (Var "z1")) ($#k17_complex1 :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k17_complex1 :::"|."::: ) (Set (Var "z2")) ($#k17_complex1 :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "y")) ($#k1_normsp_0 :::".||"::: ) ) ")" )))))) ; theorem :: CLVECT_1:107 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "holds" (Bool "(" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "iff" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" ))) ; theorem :: CLVECT_1:108 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "y")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) )))) ; theorem :: CLVECT_1:109 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "holds" (Bool (Set (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) ) ($#k9_real_1 :::"-"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "y")) ($#k1_normsp_0 :::".||"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) )))) ; theorem :: CLVECT_1:110 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x")) ($#k1_normsp_0 :::".||"::: ) ) ($#k9_real_1 :::"-"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "y")) ($#k1_normsp_0 :::".||"::: ) ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) )))) ; theorem :: CLVECT_1:111 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "w")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#k7_real_1 :::"+"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "y")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w")) ")" ) ($#k1_normsp_0 :::".||"::: ) ))))) ; theorem :: CLVECT_1:112 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; definitionlet "CNS" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); let "S" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "CNS")); let "z" be ($#m1_hidden :::"Complex":::); func "z" :::"*"::: "S" -> ($#m1_subset_1 :::"sequence":::) "of" "CNS" means :: CLVECT_1:def 14 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_normsp_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set "z" ($#k1_clvect_1 :::"*"::: ) (Set "(" "S" ($#k1_normsp_1 :::"."::: ) (Set (Var "n")) ")" )))); end; :: deftheorem defines :::"*"::: CLVECT_1:def 14 : (Bool "for" (Set (Var "CNS")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k6_clvect_1 :::"*"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b4")) ($#k1_normsp_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "S")) ($#k1_normsp_1 :::"."::: ) (Set (Var "n")) ")" )))) ")" ))))); definitionlet "CNS" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "S" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "CNS")); attr "S" is :::"convergent"::: means :: CLVECT_1:def 15 (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Point":::) "of" "CNS" "st" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" "S" ($#k1_normsp_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "g")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))))))); end; :: deftheorem defines :::"convergent"::: CLVECT_1:def 15 : (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v9_clvect_1 :::"convergent"::: ) ) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "st" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k1_normsp_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "g")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))))))) ")" ))); theorem :: CLVECT_1:113 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "S1")) "," (Set (Var "S2")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S1")) "is" ($#v9_clvect_1 :::"convergent"::: ) ) & (Bool (Set (Var "S2")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool (Set (Set (Var "S1")) ($#k2_normsp_1 :::"+"::: ) (Set (Var "S2"))) "is" ($#v9_clvect_1 :::"convergent"::: ) ))) ; theorem :: CLVECT_1:114 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "S1")) "," (Set (Var "S2")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S1")) "is" ($#v9_clvect_1 :::"convergent"::: ) ) & (Bool (Set (Var "S2")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool (Set (Set (Var "S1")) ($#k3_normsp_1 :::"-"::: ) (Set (Var "S2"))) "is" ($#v9_clvect_1 :::"convergent"::: ) ))) ; theorem :: CLVECT_1:115 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool (Set (Set (Var "S")) ($#k4_normsp_1 :::"-"::: ) (Set (Var "x"))) "is" ($#v9_clvect_1 :::"convergent"::: ) )))) ; theorem :: CLVECT_1:116 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool (Set (Set (Var "z")) ($#k6_clvect_1 :::"*"::: ) (Set (Var "S"))) "is" ($#v9_clvect_1 :::"convergent"::: ) )))) ; theorem :: CLVECT_1:117 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool (Set ($#k4_normsp_0 :::"||."::: ) (Set (Var "S")) ($#k4_normsp_0 :::".||"::: ) ) "is" ($#v2_comseq_2 :::"convergent"::: ) ))) ; definitionlet "CNS" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "S" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "CNS")); assume (Bool (Set (Const "S")) "is" ($#v9_clvect_1 :::"convergent"::: ) ) ; func :::"lim"::: "S" -> ($#m1_subset_1 :::"Point":::) "of" "CNS" means :: CLVECT_1:def 16 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" "S" ($#k1_normsp_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k5_algstr_0 :::"-"::: ) it ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))))); end; :: deftheorem defines :::"lim"::: CLVECT_1:def 16 : (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k7_clvect_1 :::"lim"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k1_normsp_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "b3")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))))) ")" )))); theorem :: CLVECT_1:118 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v9_clvect_1 :::"convergent"::: ) ) & (Bool (Set ($#k7_clvect_1 :::"lim"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set ($#k4_normsp_0 :::"||."::: ) (Set "(" (Set (Var "S")) ($#k4_normsp_1 :::"-"::: ) (Set (Var "g")) ")" ) ($#k4_normsp_0 :::".||"::: ) ) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set ($#k4_normsp_0 :::"||."::: ) (Set "(" (Set (Var "S")) ($#k4_normsp_1 :::"-"::: ) (Set (Var "g")) ")" ) ($#k4_normsp_0 :::".||"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: CLVECT_1:119 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "S1")) "," (Set (Var "S2")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S1")) "is" ($#v9_clvect_1 :::"convergent"::: ) ) & (Bool (Set (Var "S2")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool (Set ($#k7_clvect_1 :::"lim"::: ) (Set "(" (Set (Var "S1")) ($#k2_normsp_1 :::"+"::: ) (Set (Var "S2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_clvect_1 :::"lim"::: ) (Set (Var "S1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k7_clvect_1 :::"lim"::: ) (Set (Var "S2")) ")" ))))) ; theorem :: CLVECT_1:120 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "S1")) "," (Set (Var "S2")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S1")) "is" ($#v9_clvect_1 :::"convergent"::: ) ) & (Bool (Set (Var "S2")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool (Set ($#k7_clvect_1 :::"lim"::: ) (Set "(" (Set (Var "S1")) ($#k3_normsp_1 :::"-"::: ) (Set (Var "S2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_clvect_1 :::"lim"::: ) (Set (Var "S1")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k7_clvect_1 :::"lim"::: ) (Set (Var "S2")) ")" ))))) ; theorem :: CLVECT_1:121 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool (Set ($#k7_clvect_1 :::"lim"::: ) (Set "(" (Set (Var "S")) ($#k4_normsp_1 :::"-"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_clvect_1 :::"lim"::: ) (Set (Var "S")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x"))))))) ; theorem :: CLVECT_1:122 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v9_clvect_1 :::"convergent"::: ) )) "holds" (Bool (Set ($#k7_clvect_1 :::"lim"::: ) (Set "(" (Set (Var "z")) ($#k6_clvect_1 :::"*"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" ($#k7_clvect_1 :::"lim"::: ) (Set (Var "S")) ")" )))))) ; theorem :: CLVECT_1:123 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set (Var "D")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "D")) (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set (Var "D")) "," (Set (Var "d1")) "," (Set (Var "A")) "," (Set (Var "M")) "#)" ) "st" (Bool (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_clvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set (Var "V1")) ($#k2_zfmisc_1 :::":]"::: ) ))) & (Bool (Set (Var "w")) ($#r1_hidden :::"="::: ) (Set (Var "v")))) "holds" (Bool (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))))))))))))) ;