:: CONVEX4 semantic presentation begin definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; mode :::"C_Linear_Combination"::: "of" "V" -> ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) ")" ) means :: CONVEX4:def 1 (Bool "ex" (Set (Var "T")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" "V" "st" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" "V" "st" (Bool (Bool (Bool "not" (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "T"))))) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )))); end; :: deftheorem defines :::"C_Linear_Combination"::: CONVEX4:def 1 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V"))) "iff" (Bool "ex" (Set (Var "T")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "st" (Bool (Bool (Bool "not" (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "T"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )))) ")" ))); notationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "L" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) ")" ); synonym :::"Carrier"::: "L" for :::"support"::: "V"; end; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "L" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) ")" ); :: original: :::"Carrier"::: redefine func :::"Carrier"::: "L" -> ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: CONVEX4:def 2 "{" (Set (Var "v")) where v "is" ($#m1_subset_1 :::"Element":::) "of" "V" : (Bool (Set "L" ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"<>"::: ) (Set ($#k5_complex1 :::"0c"::: ) )) "}" ; end; :: deftheorem defines :::"Carrier"::: CONVEX4:def 2 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "L")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) ")" ) "holds" (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "v")) where v "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) : (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"<>"::: ) (Set ($#k5_complex1 :::"0c"::: ) )) "}" ))); registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "L" be ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Const "V")); cluster (Set ($#k13_pre_poly :::"Carrier"::: ) ) -> ($#v1_finset_1 :::"finite"::: ) ; end; theorem :: CONVEX4:1 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k5_complex1 :::"0c"::: ) )) "iff" (Bool (Bool "not" (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L"))))) ")" )))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; func :::"ZeroCLC"::: "V" -> ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" "V" means :: CONVEX4:def 3 (Bool (Set ($#k1_convex4 :::"Carrier"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )); end; :: deftheorem defines :::"ZeroCLC"::: CONVEX4:def 3 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V")))) "iff" (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))); registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; cluster (Set ($#k13_pre_poly :::"Carrier"::: ) ) -> ($#v1_xboole_0 :::"empty"::: ) ; end; theorem :: CONVEX4:2 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k5_complex1 :::"0c"::: ) )))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); mode :::"C_Linear_Combination"::: "of" "A" -> ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" "V" means :: CONVEX4:def 4 (Bool (Set ($#k1_convex4 :::"Carrier"::: ) it) ($#r1_tarski :::"c="::: ) "A"); end; :: deftheorem defines :::"C_Linear_Combination"::: CONVEX4:def 4 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "b3")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A"))) "iff" (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "b3"))) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" )))); theorem :: CONVEX4:3 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "l")) "being" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "l")) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "B")))))) ; theorem :: CONVEX4:4 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V"))) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A"))))) ; theorem :: CONVEX4:5 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "l")) "being" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set ($#k1_subset_1 :::"{}"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))) "holds" (Bool (Set (Var "l")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V")))))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "F" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "V"))); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); func "f" :::"(#)"::: "F" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") means :: CONVEX4:def 5 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "F")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set "(" "F" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set "(" "F" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"(#)"::: CONVEX4:def 5 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_convex4 :::"(#)"::: ) (Set (Var "F")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b4"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "F")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "F")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ))) ")" ) ")" ) ")" ))))); theorem :: CONVEX4:6 (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 "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "F")))) & (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k3_convex4 :::"(#)"::: ) (Set (Var "F")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))))))))) ; theorem :: CONVEX4:7 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k3_convex4 :::"(#)"::: ) (Set "(" ($#k6_finseq_1 :::"<*>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_finseq_1 :::"<*>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))))))) ; theorem :: CONVEX4:8 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k3_convex4 :::"(#)"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "v")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))) ; theorem :: CONVEX4:9 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k3_convex4 :::"(#)"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k2_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) "," (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v2")) ")" ) ($#k2_finseq_4 :::"*>"::: ) ))))) ; theorem :: CONVEX4:10 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k3_convex4 :::"(#)"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) "," (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v2")) ")" ) "," (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v3")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v3")) ")" ) ($#k3_finseq_4 :::"*>"::: ) ))))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "L" be ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Const "V")); func :::"Sum"::: "L" -> ($#m1_subset_1 :::"Element":::) "of" "V" means :: CONVEX4:def 6 (Bool "ex" (Set (Var "F")) "being" ($#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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_convex4 :::"Carrier"::: ) "L")) & (Bool it ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" "L" ($#k3_convex4 :::"(#)"::: ) (Set (Var "F")) ")" ))) ")" )); end; :: deftheorem defines :::"Sum"::: CONVEX4:def 6 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "L")))) "iff" (Bool "ex" (Set (Var "F")) "being" ($#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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L")))) & (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" (Set (Var "L")) ($#k3_convex4 :::"(#)"::: ) (Set (Var "F")) ")" ))) ")" )) ")" )))); theorem :: CONVEX4:11 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) "holds" (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set "(" ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))))) ; theorem :: CONVEX4:12 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v6_clvect_1 :::"linearly-closed"::: ) ) "iff" (Bool "for" (Set (Var "l")) "being" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A")) "holds" (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "l"))) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) ")" ))) ; theorem :: CONVEX4:13 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "l")) "being" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set ($#k1_subset_1 :::"{}"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))) "holds" (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))))) ; theorem :: CONVEX4:14 (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 "l")) "being" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) "holds" (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "l")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))))))) ; theorem :: CONVEX4:15 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2")))) "holds" (Bool "for" (Set (Var "l")) "being" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ) "holds" (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "l")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "l")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v2")) ")" )))))) ; theorem :: CONVEX4:16 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))))) ; theorem :: CONVEX4:17 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v"))))))) ; theorem :: CONVEX4:18 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) )) & (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2")))) "holds" (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v2")) ")" )))))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "L1", "L2" be ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Const "V")); :: original: :::"="::: redefine pred "L1" :::"="::: "L2" means :: CONVEX4:def 7 (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" "V" "holds" (Bool (Set "L1" ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set "L2" ($#k3_funct_2 :::"."::: ) (Set (Var "v"))))); end; :: deftheorem defines :::"="::: CONVEX4:def 7 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "L1")) ($#r1_convex4 :::"="::: ) (Set (Var "L2"))) "iff" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L1")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "L2")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))))) ")" ))); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "L1", "L2" be ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Const "V")); :: original: :::"+"::: redefine func "L1" :::"+"::: "L2" -> ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" "V" means :: CONVEX4:def 8 (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" "V" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "L1" ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k8_complex1 :::"+"::: ) (Set "(" "L2" ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" )))); end; :: deftheorem defines :::"+"::: CONVEX4:def 8 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "b4")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L1")) ($#k5_convex4 :::"+"::: ) (Set (Var "L2")))) "iff" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L1")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k8_complex1 :::"+"::: ) (Set "(" (Set (Var "L2")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" )))) ")" ))); theorem :: CONVEX4:19 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set "(" (Set (Var "L1")) ($#k5_convex4 :::"+"::: ) (Set (Var "L2")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L1")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L2")) ")" ))))) ; theorem :: CONVEX4:20 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L1")) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A"))) & (Bool (Set (Var "L2")) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A")))) "holds" (Bool (Set (Set (Var "L1")) ($#k5_convex4 :::"+"::: ) (Set (Var "L2"))) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A")))))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); let "L1", "L2" be ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Const "A")); :: original: :::"+"::: redefine func "L1" :::"+"::: "L2" -> ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" "A"; end; theorem :: CONVEX4:21 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L1")) ($#k5_convex4 :::"+"::: ) (Set (Var "L2"))) ($#r1_convex4 :::"="::: ) (Set (Set (Var "L2")) ($#k5_convex4 :::"+"::: ) (Set (Var "L1")))))) ; theorem :: CONVEX4:22 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "L3")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L1")) ($#k5_convex4 :::"+"::: ) (Set "(" (Set (Var "L2")) ($#k5_convex4 :::"+"::: ) (Set (Var "L3")) ")" )) ($#r1_convex4 :::"="::: ) (Set (Set "(" (Set (Var "L1")) ($#k5_convex4 :::"+"::: ) (Set (Var "L2")) ")" ) ($#k5_convex4 :::"+"::: ) (Set (Var "L3")))))) ; theorem :: CONVEX4:23 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L")) ($#k5_convex4 :::"+"::: ) (Set "(" ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V")) ")" )) ($#r1_convex4 :::"="::: ) (Set (Var "L"))))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "a" be ($#m1_hidden :::"Complex":::); let "L" be ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Const "V")); func "a" :::"*"::: "L" -> ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" "V" means :: CONVEX4:def 9 (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set "a" ($#k3_xcmplx_0 :::"*"::: ) (Set "(" "L" ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" )))); end; :: deftheorem defines :::"*"::: CONVEX4:def 9 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "L")) "," (Set (Var "b4")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" )))) ")" )))); theorem :: CONVEX4:24 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k5_complex1 :::"0c"::: ) ))) "holds" (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set "(" (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set (Var "L")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L"))))))) ; theorem :: CONVEX4:25 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set ($#k5_complex1 :::"0c"::: ) ) ($#k7_convex4 :::"*"::: ) (Set (Var "L"))) ($#r1_convex4 :::"="::: ) (Set ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V")))))) ; theorem :: CONVEX4:26 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A")))) "holds" (Bool (Set (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set (Var "L"))) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A"))))))) ; theorem :: CONVEX4:27 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "b")) ")" ) ($#k7_convex4 :::"*"::: ) (Set (Var "L"))) ($#r1_convex4 :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set (Var "L")) ")" ) ($#k5_convex4 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k7_convex4 :::"*"::: ) (Set (Var "L")) ")" )))))) ; theorem :: CONVEX4:28 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set "(" (Set (Var "L1")) ($#k5_convex4 :::"+"::: ) (Set (Var "L2")) ")" )) ($#r1_convex4 :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set (Var "L1")) ")" ) ($#k5_convex4 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set (Var "L2")) ")" )))))) ; theorem :: CONVEX4:29 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k7_convex4 :::"*"::: ) (Set (Var "L")) ")" )) ($#r1_convex4 :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "b")) ")" ) ($#k7_convex4 :::"*"::: ) (Set (Var "L"))))))) ; theorem :: CONVEX4:30 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set ($#k6_complex1 :::"1r"::: ) ) ($#k7_convex4 :::"*"::: ) (Set (Var "L"))) ($#r1_convex4 :::"="::: ) (Set (Var "L"))))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "L" be ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Const "V")); func :::"-"::: "L" -> ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" "V" equals :: CONVEX4:def 10 (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k7_convex4 :::"*"::: ) "L"); end; :: deftheorem defines :::"-"::: CONVEX4:def 10 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k8_convex4 :::"-"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k7_convex4 :::"*"::: ) (Set (Var "L")))))); theorem :: CONVEX4:31 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k8_convex4 :::"-"::: ) (Set (Var "L")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k10_complex1 :::"-"::: ) (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" )))))) ; theorem :: CONVEX4:32 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "L1")) ($#k5_convex4 :::"+"::: ) (Set (Var "L2"))) ($#r1_convex4 :::"="::: ) (Set ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V"))))) "holds" (Bool (Set (Var "L2")) ($#r1_convex4 :::"="::: ) (Set ($#k8_convex4 :::"-"::: ) (Set (Var "L1")))))) ; theorem :: CONVEX4:33 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k8_convex4 :::"-"::: ) (Set "(" ($#k8_convex4 :::"-"::: ) (Set (Var "L")) ")" )) ($#r1_convex4 :::"="::: ) (Set (Var "L"))))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "L1", "L2" be ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Const "V")); func "L1" :::"-"::: "L2" -> ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" "V" equals :: CONVEX4:def 11 (Set "L1" ($#k5_convex4 :::"+"::: ) (Set "(" ($#k8_convex4 :::"-"::: ) "L2" ")" )); end; :: deftheorem defines :::"-"::: CONVEX4:def 11 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L1")) ($#k9_convex4 :::"-"::: ) (Set (Var "L2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "L1")) ($#k5_convex4 :::"+"::: ) (Set "(" ($#k8_convex4 :::"-"::: ) (Set (Var "L2")) ")" ))))); theorem :: CONVEX4:34 (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 "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" (Set (Var "L1")) ($#k9_convex4 :::"-"::: ) (Set (Var "L2")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L1")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" (Set (Var "L2")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" )))))) ; theorem :: CONVEX4:35 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set "(" (Set (Var "L1")) ($#k9_convex4 :::"-"::: ) (Set (Var "L2")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L1")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L2")) ")" ))))) ; theorem :: CONVEX4:36 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L1")) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A"))) & (Bool (Set (Var "L2")) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A")))) "holds" (Bool (Set (Set (Var "L1")) ($#k9_convex4 :::"-"::: ) (Set (Var "L2"))) "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A")))))) ; theorem :: CONVEX4:37 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "L")) ($#k9_convex4 :::"-"::: ) (Set (Var "L"))) ($#r1_convex4 :::"="::: ) (Set ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V")))))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; func :::"C_LinComb"::: "V" -> ($#m1_hidden :::"set"::: ) means :: CONVEX4:def 12 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) "is" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" "V") ")" )); end; :: deftheorem defines :::"C_LinComb"::: CONVEX4:def 12 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k10_convex4 :::"C_LinComb"::: ) (Set (Var "V")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "x")) "is" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V"))) ")" )) ")" ))); registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; cluster (Set ($#k10_convex4 :::"C_LinComb"::: ) "V") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "e" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k10_convex4 :::"C_LinComb"::: ) (Set (Const "V"))); func :::"@"::: "e" -> ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" "V" equals :: CONVEX4:def 13 "e"; end; :: deftheorem defines :::"@"::: CONVEX4:def 13 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k10_convex4 :::"C_LinComb"::: ) (Set (Var "V"))) "holds" (Bool (Set ($#k11_convex4 :::"@"::: ) (Set (Var "e"))) ($#r1_hidden :::"="::: ) (Set (Var "e"))))); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "L" be ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Const "V")); func :::"@"::: "L" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k10_convex4 :::"C_LinComb"::: ) "V") equals :: CONVEX4:def 14 "L"; end; :: deftheorem defines :::"@"::: CONVEX4:def 14 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k12_convex4 :::"@"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Var "L"))))); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; func :::"C_LCAdd"::: "V" -> ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k10_convex4 :::"C_LinComb"::: ) "V" ")" ) means :: CONVEX4:def 15 (Bool "for" (Set (Var "e1")) "," (Set (Var "e2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k10_convex4 :::"C_LinComb"::: ) "V") "holds" (Bool (Set it ($#k5_binop_1 :::"."::: ) "(" (Set (Var "e1")) "," (Set (Var "e2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_convex4 :::"@"::: ) (Set (Var "e1")) ")" ) ($#k5_convex4 :::"+"::: ) (Set "(" ($#k11_convex4 :::"@"::: ) (Set (Var "e2")) ")" )))); end; :: deftheorem defines :::"C_LCAdd"::: CONVEX4:def 15 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k10_convex4 :::"C_LinComb"::: ) (Set (Var "V")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k13_convex4 :::"C_LCAdd"::: ) (Set (Var "V")))) "iff" (Bool "for" (Set (Var "e1")) "," (Set (Var "e2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k10_convex4 :::"C_LinComb"::: ) (Set (Var "V"))) "holds" (Bool (Set (Set (Var "b2")) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "e1")) "," (Set (Var "e2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_convex4 :::"@"::: ) (Set (Var "e1")) ")" ) ($#k5_convex4 :::"+"::: ) (Set "(" ($#k11_convex4 :::"@"::: ) (Set (Var "e2")) ")" )))) ")" ))); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; func :::"C_LCMult"::: "V" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set "(" ($#k10_convex4 :::"C_LinComb"::: ) "V" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "(" ($#k10_convex4 :::"C_LinComb"::: ) "V" ")" ) means :: CONVEX4:def 16 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k10_convex4 :::"C_LinComb"::: ) "V") "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "a")) "," (Set (Var "e")) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set "(" ($#k11_convex4 :::"@"::: ) (Set (Var "e")) ")" ))))); end; :: deftheorem defines :::"C_LCMult"::: CONVEX4:def 16 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_numbers :::"COMPLEX"::: ) ) "," (Set "(" ($#k10_convex4 :::"C_LinComb"::: ) (Set (Var "V")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "(" ($#k10_convex4 :::"C_LinComb"::: ) (Set (Var "V")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k14_convex4 :::"C_LCMult"::: ) (Set (Var "V")))) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k10_convex4 :::"C_LinComb"::: ) (Set (Var "V"))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "a")) "," (Set (Var "e")) ($#k4_tarski :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set "(" ($#k11_convex4 :::"@"::: ) (Set (Var "e")) ")" ))))) ")" ))); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; func :::"LC_CLSpace"::: "V" -> ($#l1_clvect_1 :::"ComplexLinearSpace":::) equals :: CONVEX4:def 17 (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k10_convex4 :::"C_LinComb"::: ) "V" ")" ) "," (Set "(" ($#k12_convex4 :::"@"::: ) (Set "(" ($#k2_convex4 :::"ZeroCLC"::: ) "V" ")" ) ")" ) "," (Set "(" ($#k13_convex4 :::"C_LCAdd"::: ) "V" ")" ) "," (Set "(" ($#k14_convex4 :::"C_LCMult"::: ) "V" ")" ) "#)" ); end; :: deftheorem defines :::"LC_CLSpace"::: CONVEX4:def 17 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) "holds" (Bool (Set ($#k15_convex4 :::"LC_CLSpace"::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#g1_clvect_1 :::"CLSStruct"::: ) "(#" (Set "(" ($#k10_convex4 :::"C_LinComb"::: ) (Set (Var "V")) ")" ) "," (Set "(" ($#k12_convex4 :::"@"::: ) (Set "(" ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V")) ")" ) ")" ) "," (Set "(" ($#k13_convex4 :::"C_LCAdd"::: ) (Set (Var "V")) ")" ) "," (Set "(" ($#k14_convex4 :::"C_LCMult"::: ) (Set (Var "V")) ")" ) "#)" ))); registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; cluster (Set ($#k15_convex4 :::"LC_CLSpace"::: ) "V") -> ($#v1_clvect_1 :::"strict"::: ) ; end; theorem :: CONVEX4:38 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k1_rlvect_2 :::"vector"::: ) "(" (Set "(" ($#k15_convex4 :::"LC_CLSpace"::: ) (Set (Var "V")) ")" ) "," (Set (Var "L1")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k1_rlvect_2 :::"vector"::: ) "(" (Set "(" ($#k15_convex4 :::"LC_CLSpace"::: ) (Set (Var "V")) ")" ) "," (Set (Var "L2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "L1")) ($#k5_convex4 :::"+"::: ) (Set (Var "L2")))))) ; theorem :: CONVEX4:39 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "a")) ($#k1_clvect_1 :::"*"::: ) (Set "(" ($#k1_rlvect_2 :::"vector"::: ) "(" (Set "(" ($#k15_convex4 :::"LC_CLSpace"::: ) (Set (Var "V")) ")" ) "," (Set (Var "L")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_convex4 :::"*"::: ) (Set (Var "L"))))))) ; theorem :: CONVEX4:40 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_rlvect_2 :::"vector"::: ) "(" (Set "(" ($#k15_convex4 :::"LC_CLSpace"::: ) (Set (Var "V")) ")" ) "," (Set (Var "L")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_convex4 :::"-"::: ) (Set (Var "L")))))) ; theorem :: CONVEX4:41 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k1_rlvect_2 :::"vector"::: ) "(" (Set "(" ($#k15_convex4 :::"LC_CLSpace"::: ) (Set (Var "V")) ")" ) "," (Set (Var "L1")) ")" ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" ($#k1_rlvect_2 :::"vector"::: ) "(" (Set "(" ($#k15_convex4 :::"LC_CLSpace"::: ) (Set (Var "V")) ")" ) "," (Set (Var "L2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "L1")) ($#k9_convex4 :::"-"::: ) (Set (Var "L2")))))) ; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func :::"LC_CLSpace"::: "A" -> ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set ($#k15_convex4 :::"LC_CLSpace"::: ) "V") means :: CONVEX4:def 18 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) "{" (Set (Var "l")) where l "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" "A" : (Bool verum) "}" ); end; :: deftheorem defines :::"LC_CLSpace"::: CONVEX4:def 18 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "b3")) "being" ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set ($#k15_convex4 :::"LC_CLSpace"::: ) (Set (Var "V"))) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k16_convex4 :::"LC_CLSpace"::: ) (Set (Var "A")))) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "l")) where l "is" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "A")) : (Bool verum) "}" ) ")" )))); begin definitionlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); let "W" be ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Const "V")); func :::"Up"::: "W" -> ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: CONVEX4:def 19 (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "W"); end; :: deftheorem defines :::"Up"::: CONVEX4:def 19 : (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k17_convex4 :::"Up"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))))); registrationlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); let "W" be ($#m1_clvect_1 :::"Subspace"::: ) "of" (Set (Const "V")); cluster (Set ($#k17_convex4 :::"Up"::: ) "W") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "S" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); attr "S" is :::"Affine"::: means :: CONVEX4:def 20 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool (Set (Var "z")) "is" ($#m1_subset_1 :::"Real":::)) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "S") & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "S")) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "z")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "y")) ")" )) ($#r2_hidden :::"in"::: ) "S"))); end; :: deftheorem defines :::"Affine"::: CONVEX4:def 20 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v1_convex4 :::"Affine"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool (Set (Var "z")) "is" ($#m1_subset_1 :::"Real":::)) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "z")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "y")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "S"))))) ")" ))); definitionlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); func :::"(Omega)."::: "V" -> ($#v1_clvect_1 :::"strict"::: ) ($#m1_clvect_1 :::"Subspace"::: ) "of" "V" equals :: CONVEX4:def 21 (Set ($#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)."::: CONVEX4:def 21 : (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#k18_convex4 :::"(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"))) "#)" ))); registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; cluster (Set ($#k2_struct_0 :::"[#]"::: ) "V") -> ($#v1_convex4 :::"Affine"::: ) ; cluster ($#v1_xboole_0 :::"empty"::: ) -> ($#v1_convex4 :::"Affine"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V")); end; registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_convex4 :::"Affine"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V")); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_convex4 :::"Affine"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V")); end; theorem :: CONVEX4:42 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "z")) "being" ($#v1_xcmplx_0 :::"complex"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_complex1 :::"Re"::: ) (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k3_complex1 :::"Re"::: ) (Set (Var "z")) ")" ))))) ; theorem :: CONVEX4:43 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "z")) "being" ($#v1_xcmplx_0 :::"complex"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_complex1 :::"Im"::: ) (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k4_complex1 :::"Im"::: ) (Set (Var "z")) ")" ))))) ; theorem :: CONVEX4:44 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "z")) "being" ($#v1_xcmplx_0 :::"complex"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_xboole_0 :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "(" (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "z")) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_real_1 :::"*"::: ) (Set ($#k17_complex1 :::"|."::: ) (Set (Var "z")) ($#k17_complex1 :::".|"::: ) ))) & (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "z")) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set ($#k17_complex1 :::"|."::: ) (Set (Var "z")) ($#k17_complex1 :::".|"::: ) ))) ")" ))) ; begin definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "M" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); let "r" be ($#m1_hidden :::"Complex":::); func "r" :::"*"::: "M" -> ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: CONVEX4:def 22 "{" (Set "(" "r" ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" ) where v "is" ($#m1_subset_1 :::"Element":::) "of" "V" : (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) "M") "}" ; end; :: deftheorem defines :::"*"::: CONVEX4:def 22 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "r")) ($#k19_convex4 :::"*"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "r")) ($#k1_clvect_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"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "M" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); attr "M" is :::"convex"::: means :: CONVEX4:def 23 (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool (Set ($#k1_xboole_0 :::"0"::: ) ) ($#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")) ($#r2_hidden :::"in"::: ) "M")) "holds" (Bool (Set (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "u")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "z")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" )) ($#r2_hidden :::"in"::: ) "M"))); end; :: deftheorem defines :::"convex"::: CONVEX4:def 23 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ) "iff" (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool (Set ($#k1_xboole_0 :::"0"::: ) ) ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "M")))) "holds" (Bool (Set (Set "(" (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set (Var "u")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "z")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "M"))))) ")" ))); theorem :: CONVEX4:45 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) )) "holds" (Bool (Set (Set (Var "z")) ($#k19_convex4 :::"*"::: ) (Set (Var "M"))) "is" ($#v2_convex4 :::"convex"::: ) )))) ; theorem :: CONVEX4:46 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ) & (Bool (Set (Var "N")) "is" ($#v2_convex4 :::"convex"::: ) )) "holds" (Bool (Set (Set (Var "M")) ($#k7_rusub_4 :::"+"::: ) (Set (Var "N"))) "is" ($#v2_convex4 :::"convex"::: ) ))) ; theorem :: CONVEX4:47 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ) & (Bool (Set (Var "N")) "is" ($#v2_convex4 :::"convex"::: ) )) "holds" (Bool (Set (Set (Var "M")) ($#k1_rusub_5 :::"-"::: ) (Set (Var "N"))) "is" ($#v2_convex4 :::"convex"::: ) ))) ; theorem :: CONVEX4:48 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ) "iff" (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool (Set ($#k1_xboole_0 :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ))) "holds" (Bool (Set (Set "(" (Set (Var "z")) ($#k19_convex4 :::"*"::: ) (Set (Var "M")) ")" ) ($#k6_rusub_4 :::"+"::: ) (Set "(" (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "z")) ")" ) ($#k19_convex4 :::"*"::: ) (Set (Var "M")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "M")))) ")" ))) ; theorem :: CONVEX4:49 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) )) "holds" (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool (Set ($#k1_xboole_0 :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "z")) ")" ) ($#k19_convex4 :::"*"::: ) (Set (Var "M")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k19_convex4 :::"*"::: ) (Set (Var "M")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "M")))))) ; theorem :: CONVEX4:50 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ) & (Bool (Set (Var "N")) "is" ($#v2_convex4 :::"convex"::: ) )) "holds" (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set "(" (Set (Var "z")) ($#k19_convex4 :::"*"::: ) (Set (Var "M")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set "(" (Set ($#k6_complex1 :::"1r"::: ) ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "z")) ")" ) ($#k19_convex4 :::"*"::: ) (Set (Var "N")) ")" )) "is" ($#v2_convex4 :::"convex"::: ) )))) ; theorem :: CONVEX4:51 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set ($#k6_complex1 :::"1r"::: ) ) ($#k19_convex4 :::"*"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "M"))))) ; theorem :: CONVEX4:52 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set ($#k5_complex1 :::"0c"::: ) ) ($#k19_convex4 :::"*"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: CONVEX4:53 (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 :: CONVEX4:54 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z1")) "," (Set (Var "z2")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "z1")) ($#k19_convex4 :::"*"::: ) (Set "(" (Set (Var "z2")) ($#k19_convex4 :::"*"::: ) (Set (Var "M")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z1")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "z2")) ")" ) ($#k19_convex4 :::"*"::: ) (Set (Var "M"))))))) ; theorem :: CONVEX4:55 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "z")) ($#k19_convex4 :::"*"::: ) (Set "(" (Set (Var "M1")) ($#k6_rusub_4 :::"+"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k19_convex4 :::"*"::: ) (Set (Var "M1")) ")" ) ($#k6_rusub_4 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k19_convex4 :::"*"::: ) (Set (Var "M2")) ")" )))))) ; theorem :: CONVEX4:56 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (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" ($#v2_convex4 :::"convex"::: ) ) "iff" (Bool (Set (Set (Var "v")) ($#k5_rusub_4 :::"+"::: ) (Set (Var "M"))) "is" ($#v2_convex4 :::"convex"::: ) ) ")" )))) ; theorem :: CONVEX4:57 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#k17_convex4 :::"Up"::: ) (Set "(" ($#k3_clvect_1 :::"(0)."::: ) (Set (Var "V")) ")" )) "is" ($#v2_convex4 :::"convex"::: ) )) ; theorem :: CONVEX4:58 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) "holds" (Bool (Set ($#k17_convex4 :::"Up"::: ) (Set "(" ($#k18_convex4 :::"(Omega)."::: ) (Set (Var "V")) ")" )) "is" ($#v2_convex4 :::"convex"::: ) )) ; theorem :: CONVEX4:59 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))) ; theorem :: CONVEX4:60 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z1")) "," (Set (Var "z2")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool (Set (Var "M1")) "is" ($#v2_convex4 :::"convex"::: ) ) & (Bool (Set (Var "M2")) "is" ($#v2_convex4 :::"convex"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "z1")) ($#k19_convex4 :::"*"::: ) (Set (Var "M1")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "z2")) ($#k19_convex4 :::"*"::: ) (Set (Var "M2")) ")" )) "is" ($#v2_convex4 :::"convex"::: ) )))) ; theorem :: CONVEX4:61 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z1")) "," (Set (Var "z2")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set "(" (Set (Var "z1")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "z2")) ")" ) ($#k19_convex4 :::"*"::: ) (Set (Var "M"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" (Set (Var "z1")) ($#k19_convex4 :::"*"::: ) (Set (Var "M")) ")" ) ($#k6_rusub_4 :::"+"::: ) (Set "(" (Set (Var "z2")) ($#k19_convex4 :::"*"::: ) (Set (Var "M")) ")" )))))) ; theorem :: CONVEX4:62 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool (Set (Var "M")) ($#r1_tarski :::"c="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "z")) ($#k19_convex4 :::"*"::: ) (Set (Var "M"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "z")) ($#k19_convex4 :::"*"::: ) (Set (Var "N"))))))) ; theorem :: CONVEX4:63 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#v1_xboole_0 :::"empty"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) "holds" (Bool (Set (Set (Var "z")) ($#k19_convex4 :::"*"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))) ; theorem :: CONVEX4:64 (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 ($#k1_xboole_0 :::"{}"::: ) ))))) ; theorem :: CONVEX4:65 (Bool "for" (Set (Var "V")) "being" ($#~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 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "M"))))) ; theorem :: CONVEX4:66 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z1")) "," (Set (Var "z2")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool "ex" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z1")) ($#r1_hidden :::"="::: ) (Set (Var "r1"))) & (Bool (Set (Var "z2")) ($#r1_hidden :::"="::: ) (Set (Var "r2"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) & (Bool (Set (Var "r2")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ")" )) & (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "z1")) ($#k19_convex4 :::"*"::: ) (Set (Var "M")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "z2")) ($#k19_convex4 :::"*"::: ) (Set (Var "M")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z1")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "z2")) ")" ) ($#k19_convex4 :::"*"::: ) (Set (Var "M"))))))) ; theorem :: CONVEX4:67 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v2_clvect_1 :::"vector-distributive"::: ) ($#v3_clvect_1 :::"scalar-distributive"::: ) ($#v4_clvect_1 :::"scalar-associative"::: ) ($#v5_clvect_1 :::"scalar-unital"::: ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "z1")) "," (Set (Var "z2")) "," (Set (Var "z3")) "being" ($#m1_hidden :::"Complex":::) "st" (Bool (Bool (Set (Var "M1")) "is" ($#v2_convex4 :::"convex"::: ) ) & (Bool (Set (Var "M2")) "is" ($#v2_convex4 :::"convex"::: ) ) & (Bool (Set (Var "M3")) "is" ($#v2_convex4 :::"convex"::: ) )) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "z1")) ($#k19_convex4 :::"*"::: ) (Set (Var "M1")) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "z2")) ($#k19_convex4 :::"*"::: ) (Set (Var "M2")) ")" ) ")" ) ($#k7_rusub_4 :::"+"::: ) (Set "(" (Set (Var "z3")) ($#k19_convex4 :::"*"::: ) (Set (Var "M3")) ")" )) "is" ($#v2_convex4 :::"convex"::: ) )))) ; theorem :: CONVEX4:68 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "F")) "being" ($#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" ($#v2_convex4 :::"convex"::: ) ) ")" )) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))) "is" ($#v2_convex4 :::"convex"::: ) ))) ; theorem :: CONVEX4:69 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v1_convex4 :::"Affine"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))) ; registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_convex4 :::"convex"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V")); end; registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; cluster ($#v1_xboole_0 :::"empty"::: ) ($#v2_convex4 :::"convex"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V")); end; theorem :: CONVEX4:70 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_csspace :::"ComplexUnitarySpace-like"::: ) ($#l1_csspace :::"CUNITSTR"::: ) (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" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set ($#k3_complex1 :::"Re"::: ) (Set "(" (Set (Var "u")) ($#k12_csspace :::".|."::: ) (Set (Var "v")) ")" )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))))) ; theorem :: CONVEX4:71 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_csspace :::"ComplexUnitarySpace-like"::: ) ($#l1_csspace :::"CUNITSTR"::: ) (Bool "for" (Set (Var "M")) "being" ($#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 ($#k3_complex1 :::"Re"::: ) (Set "(" (Set (Var "u")) ($#k12_csspace :::".|."::: ) (Set (Var "v")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))))) ; theorem :: CONVEX4:72 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_csspace :::"ComplexUnitarySpace-like"::: ) ($#l1_csspace :::"CUNITSTR"::: ) (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" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set ($#k3_complex1 :::"Re"::: ) (Set "(" (Set (Var "u")) ($#k12_csspace :::".|."::: ) (Set (Var "v")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))))) ; theorem :: CONVEX4:73 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_csspace :::"ComplexUnitarySpace-like"::: ) ($#l1_csspace :::"CUNITSTR"::: ) (Bool "for" (Set (Var "M")) "being" ($#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 ($#k3_complex1 :::"Re"::: ) (Set "(" (Set (Var "u")) ($#k12_csspace :::".|."::: ) (Set (Var "v")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))))) ; theorem :: CONVEX4:74 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_csspace :::"ComplexUnitarySpace-like"::: ) ($#l1_csspace :::"CUNITSTR"::: ) (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" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set ($#k4_complex1 :::"Im"::: ) (Set "(" (Set (Var "u")) ($#k12_csspace :::".|."::: ) (Set (Var "v")) ")" )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))))) ; theorem :: CONVEX4:75 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_csspace :::"ComplexUnitarySpace-like"::: ) ($#l1_csspace :::"CUNITSTR"::: ) (Bool "for" (Set (Var "M")) "being" ($#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 ($#k4_complex1 :::"Im"::: ) (Set "(" (Set (Var "u")) ($#k12_csspace :::".|."::: ) (Set (Var "v")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))))) ; theorem :: CONVEX4:76 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_csspace :::"ComplexUnitarySpace-like"::: ) ($#l1_csspace :::"CUNITSTR"::: ) (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" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set ($#k4_complex1 :::"Im"::: ) (Set "(" (Set (Var "u")) ($#k12_csspace :::".|."::: ) (Set (Var "v")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))))) ; theorem :: CONVEX4:77 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_csspace :::"ComplexUnitarySpace-like"::: ) ($#l1_csspace :::"CUNITSTR"::: ) (Bool "for" (Set (Var "M")) "being" ($#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 ($#k4_complex1 :::"Im"::: ) (Set "(" (Set (Var "u")) ($#k12_csspace :::".|."::: ) (Set (Var "v")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))))) ; theorem :: CONVEX4:78 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_csspace :::"ComplexUnitarySpace-like"::: ) ($#l1_csspace :::"CUNITSTR"::: ) (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" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set (Var "u")) ($#k12_csspace :::".|."::: ) (Set (Var "v")) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))))) ; theorem :: CONVEX4:79 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_csspace :::"ComplexUnitarySpace-like"::: ) ($#l1_csspace :::"CUNITSTR"::: ) (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" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set (Var "u")) ($#k12_csspace :::".|."::: ) (Set (Var "v")) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "M")) "is" ($#v2_convex4 :::"convex"::: ) ))))) ; begin definitionlet "V" be ($#l1_clvect_1 :::"ComplexLinearSpace":::); let "L" be ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Const "V")); attr "L" is :::"convex"::: means :: CONVEX4:def 24 (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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_convex4 :::"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 ($#k1_xboole_0 :::"0"::: ) )) ")" ) ")" ) ")" )) ")" )); end; :: deftheorem defines :::"convex"::: CONVEX4:def 24 : (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_convex4 :::"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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_convex4 :::"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 ($#k1_xboole_0 :::"0"::: ) )) ")" ) ")" ) ")" )) ")" )) ")" ))); theorem :: CONVEX4:80 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v3_convex4 :::"convex"::: ) )) "holds" (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: CONVEX4:81 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v3_convex4 :::"convex"::: ) ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ")" ))) "holds" (Bool "not" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L")))))))) ; theorem :: CONVEX4:82 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v3_convex4 :::"convex"::: ) )) "holds" (Bool (Set (Var "L")) ($#r1_hidden :::"<>"::: ) (Set ($#k2_convex4 :::"ZeroCLC"::: ) (Set (Var "V")))))) ; theorem :: CONVEX4:83 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v3_convex4 :::"convex"::: ) ) & (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool "(" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")))) & (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) & (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")))) ")" )))) ; theorem :: CONVEX4:84 (Bool "for" (Set (Var "V")) "being" ($#l1_clvect_1 :::"ComplexLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v3_convex4 :::"convex"::: ) ) & (Bool (Set ($#k1_convex4 :::"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 "ex" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")))) & (Bool (Set (Var "r2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")))) & (Bool (Set (Set (Var "r1")) ($#k7_real_1 :::"+"::: ) (Set (Var "r2"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) & (Bool (Set (Var "r2")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ")" )) & (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v2")) ")" ))) ")" )))) ; theorem :: CONVEX4:85 (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 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m1_convex4 :::"C_Linear_Combination"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "L")) "is" ($#v3_convex4 :::"convex"::: ) ) & (Bool (Set ($#k1_convex4 :::"Carrier"::: ) (Set (Var "L"))) ($#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")) ($#r1_hidden :::"<>"::: ) (Set (Var "v1")))) "holds" (Bool "(" (Bool "ex" (Set (Var "r1")) "," (Set (Var "r2")) "," (Set (Var "r3")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")))) & (Bool (Set (Var "r2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")))) & (Bool (Set (Var "r3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v3")))) & (Bool (Set (Set "(" (Set (Var "r1")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r2")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r3"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) & (Bool (Set (Var "r2")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) & (Bool (Set (Var "r3")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ")" )) & (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v2")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v3")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v3")) ")" ))) ")" )))) ; theorem :: CONVEX4:86 (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 "L")) "being" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) "st" (Bool (Bool (Set (Var "L")) "is" ($#v3_convex4 :::"convex"::: ) )) "holds" (Bool "(" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")))) & (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) & (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v")))) ")" )))) ; theorem :: CONVEX4:87 (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 "L")) "being" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set ($#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" ($#v3_convex4 :::"convex"::: ) )) "holds" (Bool "(" (Bool "ex" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")))) & (Bool (Set (Var "r2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) & (Bool (Set (Var "r2")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ")" )) & (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v2")) ")" ))) ")" )))) ; theorem :: CONVEX4:88 (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 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "L")) "being" ($#m2_convex4 :::"C_Linear_Combination"::: ) "of" (Set ($#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" ($#v3_convex4 :::"convex"::: ) )) "holds" (Bool "(" (Bool "ex" (Set (Var "r1")) "," (Set (Var "r2")) "," (Set (Var "r3")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")))) & (Bool (Set (Var "r2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")))) & (Bool (Set (Var "r3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v3")))) & (Bool (Set (Set "(" (Set (Var "r1")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r2")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r3"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) & (Bool (Set (Var "r2")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) & (Bool (Set (Var "r3")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ")" )) & (Bool (Set ($#k4_convex4 :::"Sum"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v2")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v2")) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "L")) ($#k3_funct_2 :::"."::: ) (Set (Var "v3")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set (Var "v3")) ")" ))) ")" )))) ; begin definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "M" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func :::"Convex-Family"::: "M" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "V" means :: CONVEX4:def 25 (Bool "for" (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" "V" "holds" (Bool "(" (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "N")) "is" ($#v2_convex4 :::"convex"::: ) ) & (Bool "M" ($#r1_tarski :::"c="::: ) (Set (Var "N"))) ")" ) ")" )); end; :: deftheorem defines :::"Convex-Family"::: CONVEX4:def 25 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (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")) ($#r1_hidden :::"="::: ) (Set ($#k20_convex4 :::"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" ($#v2_convex4 :::"convex"::: ) ) & (Bool (Set (Var "M")) ($#r1_tarski :::"c="::: ) (Set (Var "N"))) ")" ) ")" )) ")" )))); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) ; let "M" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func :::"conv"::: "M" -> ($#v2_convex4 :::"convex"::: ) ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: CONVEX4:def 26 (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k20_convex4 :::"Convex-Family"::: ) "M" ")" )); end; :: deftheorem defines :::"conv"::: CONVEX4:def 26 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k21_convex4 :::"conv"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k20_convex4 :::"Convex-Family"::: ) (Set (Var "M")) ")" ))))); theorem :: CONVEX4:89 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_clvect_1 :::"CLSStruct"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "N")) "being" ($#v2_convex4 :::"convex"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "M")) ($#r1_tarski :::"c="::: ) (Set (Var "N")))) "holds" (Bool (Set ($#k21_convex4 :::"conv"::: ) (Set (Var "M"))) ($#r1_tarski :::"c="::: ) (Set (Var "N")))))) ;