:: CONVEX1 semantic presentation 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")); let "r" be ($#m1_subset_1 :::"Real":::); func "r" :::"*"::: "M" -> ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: CONVEX1:def 1 "{" (Set "(" "r" ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" ) where v "is" ($#m1_subset_1 :::"Element":::) "of" "V" : (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) "M") "}" ; end; :: deftheorem defines :::"*"::: CONVEX1:def 1 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" ) where v "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) : (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "M"))) "}" )))); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "M" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); attr "M" is :::"convex"::: means :: CONVEX1:def 2 (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "M") & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) "M")) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "u")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" )) ($#r2_hidden :::"in"::: ) "M"))); end; :: deftheorem defines :::"convex"::: CONVEX1:def 2 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ) "iff" (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "M"))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "M")))) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "u")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "M"))))) ")" ))); theorem :: CONVEX1:1 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) )) "holds" (Bool (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "M"))) "is" ($#v1_convex1 :::"convex"::: ) )))) ; theorem :: CONVEX1:2 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ) & (Bool (Set (Var "N")) "is" ($#v1_convex1 :::"convex"::: ) )) "holds" (Bool (Set (Set (Var "M")) ($#k7_rusub_4 :::"+"::: ) (Set (Var "N"))) "is" ($#v1_convex1 :::"convex"::: ) ))) ; theorem :: CONVEX1:3 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ) & (Bool (Set (Var "N")) "is" ($#v1_convex1 :::"convex"::: ) )) "holds" (Bool (Set (Set (Var "M")) ($#k1_rusub_5 :::"-"::: ) (Set (Var "N"))) "is" ($#v1_convex1 :::"convex"::: ) ))) ; theorem :: CONVEX1:4 (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" ($#v1_convex1 :::"convex"::: ) ) "iff" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" ) ($#k6_rusub_4 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "M")))) ")" ))) ; theorem :: CONVEX1:5 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) )) "holds" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "M")))))) ; theorem :: CONVEX1:6 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ) & (Bool (Set (Var "N")) "is" ($#v1_convex1 :::"convex"::: ) )) "holds" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_convex1 :::"*"::: ) (Set (Var "N")) ")" )) "is" ($#v1_convex1 :::"convex"::: ) )))) ; theorem :: CONVEX1:7 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ) "iff" (Bool (Set (Set (Var "v")) ($#k5_rusub_4 :::"+"::: ) (Set (Var "M"))) "is" ($#v1_convex1 :::"convex"::: ) ) ")" )))) ; theorem :: CONVEX1:8 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set ($#k3_rusub_4 :::"Up"::: ) (Set "(" ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "V")) ")" )) "is" ($#v1_convex1 :::"convex"::: ) )) ; theorem :: CONVEX1:9 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set ($#k3_rusub_4 :::"Up"::: ) (Set "(" ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V")) ")" )) "is" ($#v1_convex1 :::"convex"::: ) )) ; theorem :: CONVEX1:10 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ))) ; theorem :: CONVEX1:11 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "M1")) "is" ($#v1_convex1 :::"convex"::: ) ) & (Bool (Set (Var "M2")) "is" ($#v1_convex1 :::"convex"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "r1")) ($#k1_convex1 :::"*"::: ) (Set (Var "M1")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "r2")) ($#k1_convex1 :::"*"::: ) (Set (Var "M2")) ")" )) "is" ($#v1_convex1 :::"convex"::: ) )))) ; theorem :: CONVEX1:12 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set "(" (Set (Var "r1")) ($#k7_real_1 :::"+"::: ) (Set (Var "r2")) ")" ) ($#k1_convex1 :::"*"::: ) (Set (Var "M"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" (Set (Var "r1")) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" ) ($#k6_rusub_4 :::"+"::: ) (Set "(" (Set (Var "r2")) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" )))))) ; theorem :: CONVEX1:13 (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 "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "r2")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "r1")) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "r2")) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "r1")) ($#k7_real_1 :::"+"::: ) (Set (Var "r2")) ")" ) ($#k1_convex1 :::"*"::: ) (Set (Var "M"))))))) ; theorem :: CONVEX1:14 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "," (Set (Var "r3")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "M1")) "is" ($#v1_convex1 :::"convex"::: ) ) & (Bool (Set (Var "M2")) "is" ($#v1_convex1 :::"convex"::: ) ) & (Bool (Set (Var "M3")) "is" ($#v1_convex1 :::"convex"::: ) )) "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")) ($#k1_convex1 :::"*"::: ) (Set (Var "M3")) ")" )) "is" ($#v1_convex1 :::"convex"::: ) )))) ; theorem :: CONVEX1:15 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "st" (Bool (Bool "(" "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ) ")" )) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))) "is" ($#v1_convex1 :::"convex"::: ) ))) ; theorem :: CONVEX1:16 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v2_rusub_4 :::"Affine"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ))) ; registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_convex1 :::"convex"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"))); end; registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_convex1 :::"convex"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"))); end; theorem :: CONVEX1:17 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_bhsp_1 :::"RealUnitarySpace-like"::: ) ($#l1_bhsp_1 :::"UNITSTR"::: ) (Bool "for" (Set (Var "M")) "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 "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) "{" (Set (Var "u")) where u "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set (Set (Var "u")) ($#k1_bhsp_1 :::".|."::: ) (Set (Var "v"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ))))) ; theorem :: CONVEX1:18 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_bhsp_1 :::"RealUnitarySpace-like"::: ) ($#l1_bhsp_1 :::"UNITSTR"::: ) (Bool "for" (Set (Var "M")) "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 "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) "{" (Set (Var "u")) where u "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set (Set (Var "u")) ($#k1_bhsp_1 :::".|."::: ) (Set (Var "v"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ))))) ; theorem :: CONVEX1:19 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_bhsp_1 :::"RealUnitarySpace-like"::: ) ($#l1_bhsp_1 :::"UNITSTR"::: ) (Bool "for" (Set (Var "M")) "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 "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) "{" (Set (Var "u")) where u "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set (Set (Var "u")) ($#k1_bhsp_1 :::".|."::: ) (Set (Var "v"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ))))) ; theorem :: CONVEX1:20 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_bhsp_1 :::"RealUnitarySpace-like"::: ) ($#l1_bhsp_1 :::"UNITSTR"::: ) (Bool "for" (Set (Var "M")) "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 "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) "{" (Set (Var "u")) where u "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set (Set (Var "u")) ($#k1_bhsp_1 :::".|."::: ) (Set (Var "v"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v1_convex1 :::"convex"::: ) ))))) ; begin definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "L" be ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Const "V")); attr "L" is :::"convex"::: means :: CONVEX1:def 3 (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 ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" )) ")" )); end; :: deftheorem defines :::"convex"::: CONVEX1:def 3 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_convex1 :::"convex"::: ) ) "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 ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" )) ")" )) ")" ))); theorem :: CONVEX1:21 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v2_convex1 :::"convex"::: ) )) "holds" (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: CONVEX1:22 (Bool "for" (Set (Var "V")) "being" ($#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" ($#v2_convex1 :::"convex"::: ) ) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (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 :: CONVEX1:23 (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" ($#v2_convex1 :::"convex"::: ) )) "holds" (Bool (Set (Var "L")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_rlvect_2 :::"ZeroLC"::: ) (Set (Var "V")))))) ; theorem :: CONVEX1:24 (Bool "for" (Set (Var "V")) "being" ($#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" ($#v2_convex1 :::"convex"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")))) ")" )))) ; theorem :: CONVEX1:25 (Bool "for" (Set (Var "V")) "being" ($#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")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2"))) & (Bool (Set (Var "L")) "is" ($#v2_convex1 :::"convex"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v2")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (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")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v2")) ")" ))) ")" )))) ; theorem :: CONVEX1:26 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m2_rlvect_2 :::"Linear_Combination"::: ) "of" (Set ($#k8_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) ($#k8_domain_1 :::"}"::: ) ) "st" (Bool (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2"))) & (Bool (Set (Var "v2")) ($#r1_hidden :::"<>"::: ) (Set (Var "v3"))) & (Bool (Set (Var "v3")) ($#r1_hidden :::"<>"::: ) (Set (Var "v1"))) & (Bool (Set (Var "L")) "is" ($#v2_convex1 :::"convex"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v2")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v3")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v2"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v3"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v2")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v3")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v3")) ")" ))) ")" )))) ; theorem :: CONVEX1:27 (Bool "for" (Set (Var "V")) "being" ($#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" ($#v2_convex1 :::"convex"::: ) ) & (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Num 1))))) ; theorem :: CONVEX1:28 (Bool "for" (Set (Var "V")) "being" ($#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" ($#v2_convex1 :::"convex"::: ) ) & (Bool (Set ($#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")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v2")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v2"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: CONVEX1:29 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_rlvect_2 :::"Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v2_convex1 :::"convex"::: ) ) & (Bool (Set ($#k3_rlvect_2 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k8_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) ($#k8_domain_1 :::"}"::: ) )) & (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2"))) & (Bool (Set (Var "v2")) ($#r1_hidden :::"<>"::: ) (Set (Var "v3"))) & (Bool (Set (Var "v3")) ($#r1_hidden :::"<>"::: ) (Set (Var "v1")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v2")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v3")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v2"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v3"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k6_rlvect_2 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v2")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "v3")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v3")) ")" ))) ")" )))) ; 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 :::"Convex-Family"::: "M" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "V" means :: CONVEX1:def 4 (Bool "for" (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" "V" "holds" (Bool "(" (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "N")) "is" ($#v1_convex1 :::"convex"::: ) ) & (Bool "M" ($#r1_tarski :::"c="::: ) (Set (Var "N"))) ")" ) ")" )); end; :: deftheorem defines :::"Convex-Family"::: CONVEX1:def 4 : (Bool "for" (Set (Var "V")) "being" ($#~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")) ($#r1_hidden :::"="::: ) (Set ($#k2_convex1 :::"Convex-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" ($#v1_convex1 :::"convex"::: ) ) & (Bool (Set (Var "M")) ($#r1_tarski :::"c="::: ) (Set (Var "N"))) ")" ) ")" )) ")" )))); 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 :::"conv"::: "M" -> ($#v1_convex1 :::"convex"::: ) ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: CONVEX1:def 5 (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k2_convex1 :::"Convex-Family"::: ) "M" ")" )); end; :: deftheorem defines :::"conv"::: CONVEX1:def 5 : (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 (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k2_convex1 :::"Convex-Family"::: ) (Set (Var "M")) ")" ))))); theorem :: CONVEX1:30 (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 "N")) "being" ($#v1_convex1 :::"convex"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) ($#r1_tarski :::"c="::: ) (Set (Var "N")))) "holds" (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "M"))) ($#r1_tarski :::"c="::: ) (Set (Var "N")))))) ; begin theorem :: CONVEX1:31 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x"))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "z")) "," (Set (Var "y")) ($#k11_finseq_1 :::"*>"::: ) ))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "x")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) "," (Set (Var "x")) ($#k11_finseq_1 :::"*>"::: ) ))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "x")) "," (Set (Var "y")) ($#k11_finseq_1 :::"*>"::: ) )))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "y")) "," (Set (Var "x")) ($#k11_finseq_1 :::"*>"::: ) )))) ; theorem :: CONVEX1:32 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Num 1) ($#k1_convex1 :::"*"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "M"))))) ; theorem :: CONVEX1:33 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#v1_xboole_0 :::"empty"::: ) ($#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 (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))) ; theorem :: CONVEX1:34 (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")) "holds" (Bool (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k1_convex1 :::"*"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: CONVEX1:35 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "M")) ($#k6_rusub_4 :::"+"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "M"))))) ; theorem :: CONVEX1:36 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" (Set (Var "M1")) ($#k6_rusub_4 :::"+"::: ) (Set (Var "M2")) ")" ) ($#k6_rusub_4 :::"+"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k6_rusub_4 :::"+"::: ) (Set "(" (Set (Var "M2")) ($#k6_rusub_4 :::"+"::: ) (Set (Var "M3")) ")" ))))) ; theorem :: CONVEX1:37 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set (Var "r1")) ($#k1_convex1 :::"*"::: ) (Set "(" (Set (Var "r2")) ($#k1_convex1 :::"*"::: ) (Set (Var "M")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "r1")) ($#k8_real_1 :::"*"::: ) (Set (Var "r2")) ")" ) ($#k1_convex1 :::"*"::: ) (Set (Var "M"))))))) ; theorem :: CONVEX1:38 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#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 "(" (Set (Var "M1")) ($#k6_rusub_4 :::"+"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "M1")) ")" ) ($#k6_rusub_4 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "M2")) ")" )))))) ; theorem :: CONVEX1:39 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "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")) ($#r1_tarski :::"c="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "M"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "r")) ($#k1_convex1 :::"*"::: ) (Set (Var "N"))))))) ; theorem :: CONVEX1:40 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "M")) "being" ($#v1_xboole_0 :::"empty"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "M")) ($#k6_rusub_4 :::"+"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))) ; begin registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "M", "N" be ($#v1_convex1 :::"convex"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); cluster (Set "M" ($#k3_xboole_0 :::"/\"::: ) "N") -> ($#v1_convex1 :::"convex"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "V"; end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "M" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); cluster (Set ($#k2_convex1 :::"Convex-Family"::: ) "M") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: CONVEX1:41 (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")) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "M")))))) ;