:: CIRCLED1 semantic presentation begin theorem :: CIRCLED1:1 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v3_rltopsp1 :::"circled"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "A")) ($#k1_rusub_5 :::"-"::: ) (Set (Var "B"))) "is" ($#v3_rltopsp1 :::"circled"::: ) ))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "M", "N" be ($#v3_rltopsp1 :::"circled"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); cluster (Set "M" ($#k1_rusub_5 :::"-"::: ) "N") -> ($#v3_rltopsp1 :::"circled"::: ) ; end; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "M" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); redefine attr "M" is :::"circled"::: means :: CIRCLED1:def 1 (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "M")) "holds" (Bool (Set (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) "M"))); end; :: deftheorem defines :::"circled"::: CIRCLED1:def 1 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v3_rltopsp1 :::"circled"::: ) ) "iff" (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "M")))) "holds" (Bool (Set (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) (Set (Var "M"))))) ")" ))); theorem :: CIRCLED1:2 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v3_rltopsp1 :::"circled"::: ) )) "holds" (Bool (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "M"))) "is" ($#v3_rltopsp1 :::"circled"::: ) )))) ; theorem :: CIRCLED1:3 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "M1")) "is" ($#v3_rltopsp1 :::"circled"::: ) ) & (Bool (Set (Var "M2")) "is" ($#v3_rltopsp1 :::"circled"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "r1")) ($#k1_convex1 :::"*"::: ) (Set (Var "M1")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "r2")) ($#k1_convex1 :::"*"::: ) (Set (Var "M2")) ")" )) "is" ($#v3_rltopsp1 :::"circled"::: ) )))) ; theorem :: CIRCLED1:4 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "," (Set (Var "r3")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "M1")) "is" ($#v3_rltopsp1 :::"circled"::: ) ) & (Bool (Set (Var "M2")) "is" ($#v3_rltopsp1 :::"circled"::: ) ) & (Bool (Set (Var "M3")) "is" ($#v3_rltopsp1 :::"circled"::: ) )) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "r1")) ($#k1_convex1 :::"*"::: ) (Set (Var "M1")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "r2")) ($#k1_convex1 :::"*"::: ) (Set (Var "M2")) ")" ) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "r3")) ($#k1_convex1 :::"*"::: ) (Set (Var "M3")) ")" )) "is" ($#v3_rltopsp1 :::"circled"::: ) )))) ; theorem :: CIRCLED1:5 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set ($#k3_rusub_4 :::"Up"::: ) (Set "(" ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "V")) ")" )) "is" ($#v3_rltopsp1 :::"circled"::: ) )) ; theorem :: CIRCLED1:6 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set ($#k3_rusub_4 :::"Up"::: ) (Set "(" ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V")) ")" )) "is" ($#v3_rltopsp1 :::"circled"::: ) )) ; theorem :: CIRCLED1:7 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#v3_rltopsp1 :::"circled"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "M")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "N"))) "is" ($#v3_rltopsp1 :::"circled"::: ) ))) ; theorem :: CIRCLED1:8 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#v3_rltopsp1 :::"circled"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "M")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "N"))) "is" ($#v3_rltopsp1 :::"circled"::: ) ))) ; begin definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "M" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func :::"Circled-Family"::: "M" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "V" means :: CIRCLED1:def 2 (Bool "for" (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" "V" "holds" (Bool "(" (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "N")) "is" ($#v3_rltopsp1 :::"circled"::: ) ) & (Bool "M" ($#r1_tarski :::"c="::: ) (Set (Var "N"))) ")" ) ")" )); end; :: deftheorem defines :::"Circled-Family"::: CIRCLED1:def 2 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_circled1 :::"Circled-Family"::: ) (Set (Var "M")))) "iff" (Bool "for" (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "(" (Bool (Set (Var "N")) "is" ($#v3_rltopsp1 :::"circled"::: ) ) & (Bool (Set (Var "M")) ($#r1_tarski :::"c="::: ) (Set (Var "N"))) ")" ) ")" )) ")" )))); definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "M" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func :::"Cir"::: "M" -> ($#v3_rltopsp1 :::"circled"::: ) ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: CIRCLED1:def 3 (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_circled1 :::"Circled-Family"::: ) "M" ")" )); end; :: deftheorem defines :::"Cir"::: CIRCLED1:def 3 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_circled1 :::"Cir"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_circled1 :::"Circled-Family"::: ) (Set (Var "M")) ")" ))))); registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "M" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); cluster (Set ($#k1_circled1 :::"Circled-Family"::: ) "M") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: CIRCLED1:9 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M1")) ($#r1_tarski :::"c="::: ) (Set (Var "M2")))) "holds" (Bool (Set ($#k1_circled1 :::"Circled-Family"::: ) (Set (Var "M2"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_circled1 :::"Circled-Family"::: ) (Set (Var "M1")))))) ; theorem :: CIRCLED1:10 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M1")) ($#r1_tarski :::"c="::: ) (Set (Var "M2")))) "holds" (Bool (Set ($#k2_circled1 :::"Cir"::: ) (Set (Var "M1"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_circled1 :::"Cir"::: ) (Set (Var "M2")))))) ; theorem :: CIRCLED1:11 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Var "M")) ($#r1_tarski :::"c="::: ) (Set ($#k2_circled1 :::"Cir"::: ) (Set (Var "M")))))) ; theorem :: CIRCLED1:12 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "N")) "being" ($#v3_rltopsp1 :::"circled"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) ($#r1_tarski :::"c="::: ) (Set (Var "N")))) "holds" (Bool (Set ($#k2_circled1 :::"Cir"::: ) (Set (Var "M"))) ($#r1_tarski :::"c="::: ) (Set (Var "N")))))) ; theorem :: CIRCLED1:13 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "being" ($#v3_rltopsp1 :::"circled"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_circled1 :::"Cir"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "M"))))) ; theorem :: CIRCLED1:14 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set ($#k2_circled1 :::"Cir"::: ) (Set "(" ($#k1_struct_0 :::"{}"::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: CIRCLED1:15 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set "(" ($#k2_circled1 :::"Cir"::: ) (Set (Var "M")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_circled1 :::"Cir"::: ) (Set "(" (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" )))))) ; begin definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "L" be ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Const "V")); attr "L" is :::"circled"::: means :: CIRCLED1:def 4 (Bool "ex" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "st" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k3_rlvect_2 :::"Carrier"::: ) "L")) & (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F")))) & (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set "L" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" )) ")" )); end; :: deftheorem defines :::"circled"::: CIRCLED1:def 4 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v1_circled1 :::"circled"::: ) ) "iff" (Bool "ex" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L")))) & (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F")))) & (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" )) ")" )) ")" ))); theorem :: CIRCLED1:16 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v1_circled1 :::"circled"::: ) )) "holds" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: CIRCLED1:17 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v1_circled1 :::"circled"::: ) ) & (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "not" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L")))))))) ; theorem :: CIRCLED1:18 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v1_circled1 :::"circled"::: ) )) "holds" (Bool (Set (Var "L")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_rlvect_2 :::"ZeroLC"::: ) (Set (Var "V")))))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V")) bbbadV5_RELAT_1((Set ($#k1_numbers :::"REAL"::: ) )) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) bbbadV1_VALUED_0() bbbadV2_VALUED_0() bbbadV3_VALUED_0() ($#v1_circled1 :::"circled"::: ) for ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" "V"; end; definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); mode circled_Combination of "V" is ($#v1_circled1 :::"circled"::: ) ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" "V"; end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "M" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); cluster bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V")) bbbadV5_RELAT_1((Set ($#k1_numbers :::"REAL"::: ) )) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) bbbadV1_VALUED_0() bbbadV2_VALUED_0() bbbadV3_VALUED_0() ($#v1_circled1 :::"circled"::: ) for ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" "M"; end; definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "M" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); mode circled_Combination of "M" is ($#v1_circled1 :::"circled"::: ) ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" "M"; end; definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); func :::"circledComb"::: "V" -> ($#m1_hidden :::"set"::: ) means :: CIRCLED1:def 5 (Bool "for" (Set (Var "L")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "L")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "L")) "is" ($#m1_rlvect_2 :::"circled_Combination":::) "of" "V") ")" )); end; :: deftheorem defines :::"circledComb"::: CIRCLED1:def 5 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_circled1 :::"circledComb"::: ) (Set (Var "V")))) "iff" (Bool "for" (Set (Var "L")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "L")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "L")) "is" ($#m1_rlvect_2 :::"circled_Combination":::) "of" (Set (Var "V"))) ")" )) ")" ))); definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "M" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func :::"circledComb"::: "M" -> ($#m1_hidden :::"set"::: ) means :: CIRCLED1:def 6 (Bool "for" (Set (Var "L")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "L")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "L")) "is" ($#m2_rlvect_2 :::"circled_Combination":::) "of" "M") ")" )); end; :: deftheorem defines :::"circledComb"::: CIRCLED1:def 6 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_circled1 :::"circledComb"::: ) (Set (Var "M")))) "iff" (Bool "for" (Set (Var "L")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "L")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool (Set (Var "L")) "is" ($#m2_rlvect_2 :::"circled_Combination":::) "of" (Set (Var "M"))) ")" )) ")" )))); theorem :: CIRCLED1:19 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "ex" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"circled_Combination":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Var "v"))) & (Bool "(" "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "L")) "is" ($#m2_rlvect_2 :::"circled_Combination":::) "of" (Set (Var "A"))) ")" ) ")" )))) ; theorem :: CIRCLED1:20 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2")))) "holds" (Bool "ex" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"circled_Combination":::) "of" (Set (Var "V")) "st" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "L")) "is" ($#m2_rlvect_2 :::"circled_Combination":::) "of" (Set (Var "A"))))))) ; theorem :: CIRCLED1:21 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_rlvect_2 :::"circled_Combination":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set (Var "b"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k8_rlvect_2 :::"*"::: ) (Set (Var "L1")) ")" ) ($#k7_rlvect_2 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k8_rlvect_2 :::"*"::: ) (Set (Var "L2")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_rlvect_2 :::"Carrier"::: ) (Set "(" (Set (Var "a")) ($#k8_rlvect_2 :::"*"::: ) (Set (Var "L1")) ")" ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k3_rlvect_2 :::"Carrier"::: ) (Set "(" (Set (Var "b")) ($#k8_rlvect_2 :::"*"::: ) (Set (Var "L2")) ")" ) ")" )))))) ; theorem :: CIRCLED1:22 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v1_circled1 :::"circled"::: ) ) & (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")))) ")" )))) ; theorem :: CIRCLED1:23 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v1_circled1 :::"circled"::: ) ) & (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) )) & (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v2")) ")" ))) ")" )))) ; theorem :: CIRCLED1:24 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) "st" (Bool (Bool (Set (Var "L")) "is" ($#v1_circled1 :::"circled"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")))) ")" )))) ; theorem :: CIRCLED1:25 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ) "st" (Bool (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2"))) & (Bool (Set (Var "L")) "is" ($#v1_circled1 :::"circled"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v2")) ")" ))) ")" )))) ;