:: SIMPLEX1 semantic presentation begin theorem :: SIMPLEX1:1 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) (Bool "for" (Set (Var "C")) "being" ($#m1_hidden :::"Cardinal":::) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k9_relat_1 :::"Im"::: ) "(" (Set (Var "R")) "," (Set (Var "x")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "C"))) ")" )) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "R"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "R")) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "R")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) ($#k1_card_2 :::"+`"::: ) (Set "(" (Set (Var "C")) ($#k2_card_2 :::"*`"::: ) (Set "(" ($#k1_card_1 :::"card"::: ) (Set (Var "X")) ")" ) ")" )))))) ; theorem :: SIMPLEX1:2 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Y")) ")" ) ($#k3_real_1 :::"+"::: ) (Num 1)))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) )) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ))))) ; definitionlet "X" be ($#l1_struct_0 :::"1-sorted"::: ) ; mode SimplicialComplexStr of "X" is ($#m1_simplex0 :::"SimplicialComplexStr"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X"); mode SimplicialComplex of "X" is ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X"); end; definitionlet "X" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "K" be ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "X")); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "K")); func :::"@"::: "A" -> ($#m1_subset_1 :::"Subset":::) "of" "X" equals :: SIMPLEX1:def 1 "A"; end; :: deftheorem defines :::"@"::: SIMPLEX1:def 1 : (Bool "for" (Set (Var "X")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "K")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k1_simplex1 :::"@"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A")))))); definitionlet "X" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "K" be ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "X")); let "A" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "K")); func :::"@"::: "A" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "X" equals :: SIMPLEX1:def 2 "A"; end; :: deftheorem defines :::"@"::: SIMPLEX1:def 2 : (Bool "for" (Set (Var "X")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "K")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k2_simplex1 :::"@"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A")))))); theorem :: SIMPLEX1:3 (Bool "for" (Set (Var "X")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "K")) "being" ($#v1_matroid0 :::"subset-closed"::: ) ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v6_simplex0 :::"total"::: ) )) "holds" (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v3_pre_topc :::"simplex-like"::: ) )) "holds" (Bool (Set ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "S")) ")" ) ($#k6_domain_1 :::"}"::: ) )) "is" ($#m2_simplex0 :::"SubSimplicialComplex"::: ) "of" (Set (Var "K")))))) ; begin definitionlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "Kr" be ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "RLS")); func :::"|.":::"Kr":::".|"::: -> ($#m1_subset_1 :::"Subset":::) "of" "RLS" means :: SIMPLEX1:def 3 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "Kr" "st" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "A")) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"|."::: SIMPLEX1:def 3 : (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "Kr")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "RLS")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kr")) ($#k3_simplex1 :::".|"::: ) )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Kr")) "st" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "A")) ")" ))) ")" )) ")" )) ")" )))); theorem :: SIMPLEX1:4 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "K1r")) "," (Set (Var "K2r")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "RLS")) "st" (Bool (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "K1r"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "K2r"))))) "holds" (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "K1r")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "K2r")) ($#k3_simplex1 :::".|"::: ) )))) ; theorem :: SIMPLEX1:5 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "Kr")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "RLS")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Kr")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) )) "holds" (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kr")) ($#k3_simplex1 :::".|"::: ) ))))) ; theorem :: SIMPLEX1:6 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "K")) "being" ($#v1_matroid0 :::"subset-closed"::: ) ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "K")) ($#k3_simplex1 :::".|"::: ) )) "iff" (Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) "st" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rlaffin2 :::"Int"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "A")) ")" ))) ")" )) ")" )))) ; theorem :: SIMPLEX1:7 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "Kr")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "RLS")) "holds" (Bool "(" (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kr")) ($#k3_simplex1 :::".|"::: ) ) "is" ($#v1_xboole_0 :::"empty"::: ) ) "iff" (Bool (Set (Var "Kr")) "is" ($#v4_simplex0 :::"empty-membered"::: ) ) ")" ))) ; theorem :: SIMPLEX1:8 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RLS")) "holds" (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "A")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k3_simplex1 :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "A")))))) ; theorem :: SIMPLEX1:9 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "RLS")) "holds" (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ) ")" ) ($#k3_simplex1 :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_simplex1 :::"|."::: ) (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set (Var "A")) ")" ) ($#k3_simplex1 :::".|"::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k3_simplex1 :::"|."::: ) (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set (Var "B")) ")" ) ($#k3_simplex1 :::".|"::: ) ))))) ; begin definitionlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "Kr" be ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "RLS")); mode :::"SubdivisionStr"::: "of" "Kr" -> ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" "RLS" means :: SIMPLEX1:def 4 (Bool "(" (Bool (Set ($#k3_simplex1 :::"|."::: ) "Kr" ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_simplex1 :::"|."::: ) it ($#k3_simplex1 :::".|"::: ) )) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" it "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) )) "holds" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" "Kr" "st" (Bool "(" (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "B")) ")" ))) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"SubdivisionStr"::: SIMPLEX1:def 4 : (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "Kr")) "," (Set (Var "b3")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "RLS")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" (Set (Var "Kr"))) "iff" (Bool "(" (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kr")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "b3")) ($#k3_simplex1 :::".|"::: ) )) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "b3")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) )) "holds" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Kr")) "st" (Bool "(" (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "B")) ")" ))) ")" )) ")" ) ")" ) ")" ))); theorem :: SIMPLEX1:10 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "Kr")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "RLS")) (Bool "for" (Set (Var "P")) "being" ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" (Set (Var "Kr")) "holds" (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kr")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "P")) ($#k3_simplex1 :::".|"::: ) ))))) ; registrationlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "Kr" be ($#v4_simplex0 :::"with_non-empty_element"::: ) ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "RLS")); cluster -> ($#v4_simplex0 :::"with_non-empty_element"::: ) for ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" "Kr"; end; theorem :: SIMPLEX1:11 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "Kr")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "RLS")) "holds" (Bool (Set (Var "Kr")) "is" ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" (Set (Var "Kr"))))) ; theorem :: SIMPLEX1:12 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "Kr")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "RLS")) "holds" (Bool (Set ($#k5_simplex0 :::"Complex_of"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Kr")))) "is" ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" (Set (Var "Kr"))))) ; theorem :: SIMPLEX1:13 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "K")) "being" ($#v1_matroid0 :::"subset-closed"::: ) ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "SF")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "SF")) ($#r1_hidden :::"="::: ) (Set ($#k2_cohsp_1 :::"Sub_of_Fin"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "K")))))) "holds" (Bool (Set ($#k5_simplex0 :::"Complex_of"::: ) (Set (Var "SF"))) "is" ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" (Set (Var "K")))))) ; theorem :: SIMPLEX1:14 (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "Kr")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "RLS")) (Bool "for" (Set (Var "P1")) "being" ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" (Set (Var "Kr")) (Bool "for" (Set (Var "P2")) "being" ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" (Set (Var "P1")) "holds" (Bool (Set (Var "P2")) "is" ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" (Set (Var "Kr"))))))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "K" be ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "V")); cluster ($#v1_matroid0 :::"subset-closed"::: ) ($#v3_matroid0 :::"finite-membered"::: ) for ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" "K"; end; definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "K" be ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "V")); mode Subdivision of "K" is ($#v1_matroid0 :::"subset-closed"::: ) ($#v3_matroid0 :::"finite-membered"::: ) ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" "K"; end; theorem :: SIMPLEX1:15 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "K")) "being" ($#v5_simplex0 :::"with_empty_element"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "K")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "K"))))) "holds" (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_orders_1 :::"BOOL"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool (Bool "(" "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Simplex":::) "of" (Set (Var "K")) "st" (Bool (Bool (Bool "not" (Set (Var "S")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool (Set (Set (Var "B")) ($#k1_funct_1 :::"."::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "S")) ")" ))) ")" )) "holds" (Bool (Set ($#k10_simplex0 :::"subdivision"::: ) "(" (Set (Var "B")) "," (Set (Var "K")) ")" ) "is" ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" (Set (Var "K")))))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "Kv" be ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Const "V")); cluster ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_matroid0 :::"subset-closed"::: ) ($#v3_matroid0 :::"finite-membered"::: ) for ($#m1_simplex1 :::"SubdivisionStr"::: ) "of" "Kv"; end; begin definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "Kv" be ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Const "V")); assume (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Const "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Const "Kv")))) ; func :::"BCS"::: "Kv" -> ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex1 :::"Subdivision":::) "of" "Kv" equals :: SIMPLEX1:def 5 (Set ($#k10_simplex0 :::"subdivision"::: ) "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) "V" ")" ) "," "Kv" ")" ); end; :: deftheorem defines :::"BCS"::: SIMPLEX1:def 5 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv"))))) "holds" (Bool (Set ($#k4_simplex1 :::"BCS"::: ) (Set (Var "Kv"))) ($#r1_hidden :::"="::: ) (Set ($#k10_simplex0 :::"subdivision"::: ) "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) "," (Set (Var "Kv")) ")" )))); definitionlet "n" be ($#m1_hidden :::"Nat":::); let "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "Kv" be ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Const "V")); assume (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Const "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Const "Kv")))) ; func :::"BCS"::: "(" "n" "," "Kv" ")" -> ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex1 :::"Subdivision":::) "of" "Kv" equals :: SIMPLEX1:def 6 (Set ($#k11_simplex0 :::"subdivision"::: ) "(" "n" "," (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) "V" ")" ) "," "Kv" ")" ); end; :: deftheorem defines :::"BCS"::: SIMPLEX1:def 6 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv"))))) "holds" (Bool (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "n")) "," (Set (Var "Kv")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_simplex0 :::"subdivision"::: ) "(" (Set (Var "n")) "," (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) "," (Set (Var "Kv")) ")" ))))); theorem :: SIMPLEX1:16 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv"))))) "holds" (Bool (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Set ($#k1_xboole_0 :::"0"::: ) ) "," (Set (Var "Kv")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "Kv"))))) ; theorem :: SIMPLEX1:17 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv"))))) "holds" (Bool (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Num 1) "," (Set (Var "Kv")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_simplex1 :::"BCS"::: ) (Set (Var "Kv")))))) ; theorem :: SIMPLEX1:18 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv"))))) "holds" (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set "(" ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "n")) "," (Set (Var "Kv")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv"))))))) ; theorem :: SIMPLEX1:19 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv"))))) "holds" (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set "(" ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "n")) "," (Set (Var "Kv")) ")" ")" ) ($#k3_simplex1 :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ))))) ; theorem :: SIMPLEX1:20 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv"))))) "holds" (Bool (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Set "(" (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "Kv")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_simplex1 :::"BCS"::: ) (Set "(" ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "n")) "," (Set (Var "Kv")) ")" ")" )))))) ; theorem :: SIMPLEX1:21 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv")))) & (Bool (Set ($#k6_simplex0 :::"degree"::: ) (Set (Var "Kv"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_xboole_0 :::"0"::: ) ))) "holds" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Kv"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Kv"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#k4_simplex1 :::"BCS"::: ) (Set (Var "Kv")))))) ; theorem :: SIMPLEX1:22 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) & (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv")))) & (Bool (Set ($#k6_simplex0 :::"degree"::: ) (Set (Var "Kv"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_xboole_0 :::"0"::: ) ))) "holds" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Kv"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Kv"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "n")) "," (Set (Var "Kv")) ")" ))))) ; theorem :: SIMPLEX1:23 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "Sv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m2_simplex0 :::"SubSimplicialComplex"::: ) "of" (Set (Var "Kv")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv")))) & (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Sv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Sv"))))) "holds" (Bool (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "n")) "," (Set (Var "Sv")) ")" ) "is" ($#m2_simplex0 :::"SubSimplicialComplex"::: ) "of" (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "n")) "," (Set (Var "Kv")) ")" )))))) ; theorem :: SIMPLEX1:24 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv"))))) "holds" (Bool (Set ($#k4_simplex0 :::"Vertices"::: ) (Set (Var "Kv"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_simplex0 :::"Vertices"::: ) (Set "(" ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "n")) "," (Set (Var "Kv")) ")" ")" )))))) ; registrationlet "n" be ($#m1_hidden :::"Nat":::); let "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "K" be ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v6_simplex0 :::"total"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Const "V")); cluster (Set ($#k5_simplex1 :::"BCS"::: ) "(" "n" "," "K" ")" ) -> ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v6_simplex0 :::"total"::: ) ; end; registrationlet "n" be ($#m1_hidden :::"Nat":::); let "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "K" be ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v2_simplex0 :::"finite-vertices"::: ) ($#v6_simplex0 :::"total"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Const "V")); cluster (Set ($#k5_simplex1 :::"BCS"::: ) "(" "n" "," "K" ")" ) -> ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v2_simplex0 :::"finite-vertices"::: ) ; end; begin definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "K" be ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "V")); attr "K" is :::"affinely-independent"::: means :: SIMPLEX1:def 7 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "K" "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) )) "holds" (Bool (Set ($#k1_simplex1 :::"@"::: ) (Set (Var "A"))) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) )); end; :: deftheorem defines :::"affinely-independent"::: SIMPLEX1:def 7 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "K")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "K")) "is" ($#v1_simplex1 :::"affinely-independent"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) )) "holds" (Bool (Set ($#k1_simplex1 :::"@"::: ) (Set (Var "A"))) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) )) ")" ))); definitionlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; let "Kr" be ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "RLS")); attr "Kr" is :::"simplex-join-closed"::: means :: SIMPLEX1:def 8 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" "Kr" "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"simplex-like"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "B")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" ) ")" )))); end; :: deftheorem defines :::"simplex-join-closed"::: SIMPLEX1:def 8 : (Bool "for" (Set (Var "RLS")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) (Bool "for" (Set (Var "Kr")) "being" ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "RLS")) "holds" (Bool "(" (Bool (Set (Var "Kr")) "is" ($#v2_simplex1 :::"simplex-join-closed"::: ) ) "iff" (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Kr")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"simplex-like"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "A")) ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "B")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" ) ")" )))) ")" ))); registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster ($#v4_simplex0 :::"empty-membered"::: ) -> ($#v1_simplex1 :::"affinely-independent"::: ) for ($#m1_simplex0 :::"SimplicialComplexStr"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"); let "F" be ($#v2_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "V")); cluster (Set ($#k5_simplex0 :::"Complex_of"::: ) "F") -> ($#v1_simplex1 :::"affinely-independent"::: ) ; end; registrationlet "RLS" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_rlvect_1 :::"RLSStruct"::: ) ; cluster ($#v4_simplex0 :::"empty-membered"::: ) -> ($#v2_simplex1 :::"simplex-join-closed"::: ) for ($#m1_simplex0 :::"SimplicialComplexStr"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "RLS"); end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "I" be ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); cluster (Set ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) "I" ($#k6_domain_1 :::"}"::: ) )) -> ($#v2_simplex1 :::"simplex-join-closed"::: ) ; end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster (Num 1) ($#v3_card_1 :::"-element"::: ) ($#v1_rlaffin1 :::"affinely-independent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V")); end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster ($#v1_matroid0 :::"subset-closed"::: ) ($#v3_matroid0 :::"finite-membered"::: ) ($#v2_simplex0 :::"finite-vertices"::: ) ($#v4_simplex0 :::"with_non-empty_element"::: ) ($#v6_simplex0 :::"total"::: ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) for ($#m1_simplex0 :::"SimplicialComplexStr"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"); end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "K" be ($#v1_simplex1 :::"affinely-independent"::: ) ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "V")); cluster -> ($#v1_simplex1 :::"affinely-independent"::: ) for ($#m2_simplex0 :::"SubSimplicialComplex"::: ) "of" "K"; end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "K" be ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Const "V")); cluster -> ($#v2_simplex1 :::"simplex-join-closed"::: ) for ($#m2_simplex0 :::"SubSimplicialComplex"::: ) "of" "K"; end; theorem :: SIMPLEX1:25 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "K")) "being" ($#v1_matroid0 :::"subset-closed"::: ) ($#m1_simplex0 :::"SimplicialComplexStr":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "K")) "is" ($#v2_simplex1 :::"simplex-join-closed"::: ) ) "iff" (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set ($#k1_rlaffin2 :::"Int"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "A")) ")" )) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rlaffin2 :::"Int"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "B")) ")" )))) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) ")" ))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "Ka" be ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Const "V")); let "S" be ($#m1_subset_1 :::"Simplex":::) "of" (Set (Const "Ka")); cluster (Set ($#k1_simplex1 :::"@"::: ) "S") -> ($#v1_rlaffin1 :::"affinely-independent"::: ) ; end; theorem :: SIMPLEX1:26 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Ks")) "being" ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "As")) "," (Set (Var "Bs")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Ks")) "st" (Bool (Bool (Set (Var "As")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set (Var "Bs")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set ($#k1_rlaffin2 :::"Int"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "As")) ")" )) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "Bs")) ")" )))) "holds" (Bool (Set (Var "As")) ($#r1_tarski :::"c="::: ) (Set (Var "Bs")))))) ; theorem :: SIMPLEX1:27 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Ks")) "being" ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "As")) "," (Set (Var "Bs")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Ks")) "st" (Bool (Bool (Set (Var "As")) "is" ($#v3_pre_topc :::"simplex-like"::: ) ) & (Bool (Set ($#k1_simplex1 :::"@"::: ) (Set (Var "As"))) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) & (Bool (Set (Var "Bs")) "is" ($#v3_pre_topc :::"simplex-like"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_rlaffin2 :::"Int"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "As")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "Bs")) ")" ))) "iff" (Bool (Set (Var "As")) ($#r1_tarski :::"c="::: ) (Set (Var "Bs"))) ")" )))) ; theorem :: SIMPLEX1:28 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Ka")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Ka")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Ka"))))) "holds" (Bool (Set ($#k4_simplex1 :::"BCS"::: ) (Set (Var "Ka"))) "is" ($#v1_simplex1 :::"affinely-independent"::: ) ))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "Ka" be ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v6_simplex0 :::"total"::: ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Const "V")); cluster (Set ($#k4_simplex1 :::"BCS"::: ) "Ka") -> ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_simplex1 :::"affinely-independent"::: ) ; let "n" be ($#m1_hidden :::"Nat":::); cluster (Set ($#k5_simplex1 :::"BCS"::: ) "(" "n" "," "Ka" ")" ) -> ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_simplex1 :::"affinely-independent"::: ) ; end; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "Kas" be ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Const "V")); cluster (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) "V" ")" ) ($#k5_relat_1 :::"|"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" "Kas")) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; theorem :: SIMPLEX1:29 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kas")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kas")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kas"))))) "holds" (Bool (Set ($#k4_simplex1 :::"BCS"::: ) (Set (Var "Kas"))) "is" ($#v2_simplex1 :::"simplex-join-closed"::: ) ))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "K" be ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v6_simplex0 :::"total"::: ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Const "V")); cluster (Set ($#k4_simplex1 :::"BCS"::: ) "K") -> ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ; let "n" be ($#m1_hidden :::"Nat":::); cluster (Set ($#k5_simplex1 :::"BCS"::: ) "(" "n" "," "K" ")" ) -> ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ; end; theorem :: SIMPLEX1:30 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kv")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kv")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kv")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_simplex0 :::"degree"::: ) (Set (Var "Kv"))))) "holds" (Bool "ex" (Set (Var "S")) "being" ($#m1_subset_1 :::"Simplex":::) "of" (Set (Var "Kv")) "st" (Bool "(" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1))) & (Bool (Set ($#k1_simplex1 :::"@"::: ) (Set (Var "S"))) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) ")" )) ")" )) "holds" (Bool (Set ($#k6_simplex0 :::"degree"::: ) (Set (Var "Kv"))) ($#r1_hidden :::"="::: ) (Set ($#k6_simplex0 :::"degree"::: ) (Set "(" ($#k4_simplex1 :::"BCS"::: ) (Set (Var "Kv")) ")" ))))) ; theorem :: SIMPLEX1:31 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Ka")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Ka")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Ka"))))) "holds" (Bool (Set ($#k6_simplex0 :::"degree"::: ) (Set (Var "Ka"))) ($#r1_hidden :::"="::: ) (Set ($#k6_simplex0 :::"degree"::: ) (Set "(" ($#k4_simplex1 :::"BCS"::: ) (Set (Var "Ka")) ")" ))))) ; theorem :: SIMPLEX1:32 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Ka")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Ka")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Ka"))))) "holds" (Bool (Set ($#k6_simplex0 :::"degree"::: ) (Set (Var "Ka"))) ($#r1_hidden :::"="::: ) (Set ($#k6_simplex0 :::"degree"::: ) (Set "(" ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "n")) "," (Set (Var "Ka")) ")" ")" )))))) ; theorem :: SIMPLEX1:33 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kas")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "S")) "being" ($#v1_tops_2 :::"simplex-like"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "Kas")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) )) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "S")) ")" )))))) ; theorem :: SIMPLEX1:34 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Kas")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "S1")) "," (Set (Var "S2")) "being" ($#v1_tops_2 :::"simplex-like"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "Kas")) "st" (Bool (Bool (Set ($#k3_simplex1 :::"|."::: ) (Set (Var "Kas")) ($#k3_simplex1 :::".|"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "Kas")))) & (Bool (Set (Var "S1")) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "S2"))) "is" ($#m1_subset_1 :::"Simplex":::) "of" (Set "(" ($#k4_simplex1 :::"BCS"::: ) (Set (Var "Kas")) ")" )) & (Bool (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "S1"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "S2"))))) "holds" (Bool "(" (Bool (Set (Var "S1")) ($#r1_tarski :::"c="::: ) (Set (Var "S2"))) & (Bool (Set (Var "S2")) "is" ($#v6_ordinal1 :::"c=-linear"::: ) ) ")" )))) ; theorem :: SIMPLEX1:35 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Aff")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "Bf")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "Aff"))) & (Bool (Set (Set "(" (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "S")) ")" ) ($#k1_xxreal_3 :::"+"::: ) (Set (Var "n")) ")" ) ($#k3_real_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Aff"))))) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "Bf")) "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Set (Var "n")) ($#k1_xxreal_3 :::"+"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "S")) ")" )) "," (Set ($#k4_simplex1 :::"BCS"::: ) (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ))) & (Bool (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "Bf"))) ")" ) "iff" (Bool "ex" (Set (Var "T")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "T")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "S"))) & (Bool (Set (Set (Var "T")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "S"))) "is" ($#v6_ordinal1 :::"c=-linear"::: ) ) & (Bool (Set (Set (Var "T")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "S"))) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1))) & (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "T"))) ($#r1_tarski :::"c="::: ) (Set (Var "Aff"))) & (Bool (Set (Var "Bf")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "S")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "T")) ")" ))) ")" )) ")" )))))) ; theorem :: SIMPLEX1:36 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Aff")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "Sf")) "being" ($#v6_ordinal1 :::"c=-linear"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v5_finset_1 :::"finite-membered"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "Sf")) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf"))) ($#r1_tarski :::"c="::: ) (Set (Var "Aff")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Sf"))) "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf")) ")" ) ")" ) ($#k5_real_1 :::"-"::: ) (Num 1)) "," (Set ($#k4_simplex1 :::"BCS"::: ) (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k1_xboole_0 :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf")) ")" )))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Sf"))) & (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" ))) ")" )))) ; theorem :: SIMPLEX1:37 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "S")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v6_ordinal1 :::"c=-linear"::: ) ) & (Bool (Set (Var "S")) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "S")) ")" )))) "holds" (Bool "for" (Set (Var "Af")) "," (Set (Var "Bf")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Bool "not" (Set (Var "Af")) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set (Var "Af")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "S")))) & (Bool (Set (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "S")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Af"))) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) ) & (Bool (Set (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "S")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Af"))) ($#r1_tarski :::"c="::: ) (Set (Var "Bf")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "S")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "S")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Af")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" )) "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set ($#k5_card_1 :::"card"::: ) (Set (Var "S"))) "," (Set ($#k4_simplex1 :::"BCS"::: ) (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Bf")) ($#k6_domain_1 :::"}"::: ) ) ")" )))))) ; theorem :: SIMPLEX1:38 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Sf")) "being" ($#v6_ordinal1 :::"c=-linear"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v5_finset_1 :::"finite-membered"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "Sf")) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Sf"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf")) ")" )))) "holds" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "st" (Bool (Bool (Bool "not" (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf"))))) & (Bool (Set (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) )) "holds" (Bool "{" (Set (Var "S1")) where S1 "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Sf"))) "," (Set ($#k4_simplex1 :::"BCS"::: ) (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" )) : (Bool (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Sf"))) ($#r1_tarski :::"c="::: ) (Set (Var "S1"))) "}" ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Sf")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) ($#k6_domain_1 :::"}"::: ) ))))) ; theorem :: SIMPLEX1:39 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Sf")) "being" ($#v6_ordinal1 :::"c=-linear"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v5_finset_1 :::"finite-membered"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "Sf")) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Sf")) ")" ) ($#k3_real_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf")) ")" ))) & (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf"))) "is" ($#v1_rlaffin1 :::"affinely-independent"::: ) )) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "S1")) where S1 "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Sf"))) "," (Set ($#k4_simplex1 :::"BCS"::: ) (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sf")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" )) : (Bool (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Sf"))) ($#r1_tarski :::"c="::: ) (Set (Var "S1"))) "}" ) ($#r1_hidden :::"="::: ) (Num 2)))) ; theorem :: SIMPLEX1:40 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "K")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v6_simplex0 :::"total"::: ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "Aff")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "Aff")) "is" ($#m1_subset_1 :::"Simplex":::) "of" (Set (Var "K")))) "holds" (Bool "(" (Bool (Set (Var "B")) "is" ($#m1_subset_1 :::"Simplex":::) "of" (Set "(" ($#k4_simplex1 :::"BCS"::: ) (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" )) "iff" (Bool "(" (Bool (Set (Var "B")) "is" ($#m1_subset_1 :::"Simplex":::) "of" (Set "(" ($#k4_simplex1 :::"BCS"::: ) (Set (Var "K")) ")" )) & (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "B"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "Aff")))) ")" ) ")" ))))) ; theorem :: SIMPLEX1:41 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "K")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v6_simplex0 :::"total"::: ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "Af")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "Sk")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_tops_2 :::"simplex-like"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "Sk")) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Sk")) ")" ) ($#k1_xxreal_3 :::"+"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_simplex0 :::"degree"::: ) (Set (Var "K"))))) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "Af")) "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Set (Var "n")) ($#k1_xxreal_3 :::"+"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Sk")) ")" )) "," (Set ($#k4_simplex1 :::"BCS"::: ) (Set (Var "K")))) & (Bool (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Sk"))) ($#r1_tarski :::"c="::: ) (Set (Var "Af"))) ")" ) "iff" (Bool "ex" (Set (Var "Tk")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_tops_2 :::"simplex-like"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "K")) "st" (Bool "(" (Bool (Set (Var "Tk")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "Sk"))) & (Bool (Set (Set (Var "Tk")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Sk"))) "is" ($#v6_ordinal1 :::"c=-linear"::: ) ) & (Bool (Set (Set (Var "Tk")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Sk"))) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Tk"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "Af")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Sk")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Tk")) ")" ))) ")" )) ")" )))))) ; theorem :: SIMPLEX1:42 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "K")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v6_simplex0 :::"total"::: ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "Sk")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_tops_2 :::"simplex-like"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "Ak")) "being" ($#m1_subset_1 :::"Simplex":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "Sk")) "is" ($#v6_ordinal1 :::"c=-linear"::: ) ) & (Bool (Set (Var "Sk")) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Sk"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sk")) ")" ))) & (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sk"))) ($#r1_tarski :::"c="::: ) (Set (Var "Ak"))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Ak"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Sk")) ")" ) ($#k3_real_1 :::"+"::: ) (Num 1)))) "holds" (Bool "{" (Set (Var "S1")) where S1 "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Sk"))) "," (Set ($#k4_simplex1 :::"BCS"::: ) (Set (Var "K"))) : (Bool "(" (Bool (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Sk"))) ($#r1_tarski :::"c="::: ) (Set (Var "S1"))) & (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "S1")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "Ak")) ")" ))) ")" ) "}" ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Sk")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Ak")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) ($#k6_domain_1 :::"}"::: ) )))))) ; theorem :: SIMPLEX1:43 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "K")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v6_simplex0 :::"total"::: ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "Sk")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_tops_2 :::"simplex-like"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "Sk")) "is" ($#v6_ordinal1 :::"c=-linear"::: ) ) & (Bool (Set (Var "Sk")) "is" ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ) & (Bool (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Sk")) ")" ) ($#k3_real_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sk")) ")" )))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "S1")) where S1 "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Sk"))) "," (Set ($#k4_simplex1 :::"BCS"::: ) (Set (Var "K"))) : (Bool "(" (Bool (Set (Set "(" ($#k2_rlaffin2 :::"center_of_mass"::: ) (Set (Var "V")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Sk"))) ($#r1_tarski :::"c="::: ) (Set (Var "S1"))) & (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "S1")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "Sk")) ")" ) ")" ))) ")" ) "}" ) ($#r1_hidden :::"="::: ) (Num 2))))) ; theorem :: SIMPLEX1:44 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "K")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v6_simplex0 :::"total"::: ) ($#v1_simplex1 :::"affinely-independent"::: ) ($#v2_simplex1 :::"simplex-join-closed"::: ) ($#m1_simplex0 :::"SimplicialComplex":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "Af")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "K")) "is" ($#m1_simplex1 :::"Subdivision":::) "of" (Set ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Af")) ($#k6_domain_1 :::"}"::: ) ))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Af"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1))) & (Bool (Set ($#k6_simplex0 :::"degree"::: ) (Set (Var "K"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "S")) "being" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Set (Var "n")) ($#k5_real_1 :::"-"::: ) (Num 1)) "," (Set (Var "K")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set (Var "S1")) where S1 "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Var "n")) "," (Set (Var "K")) : (Bool (Set (Var "S")) ($#r1_tarski :::"c="::: ) (Set (Var "S1"))) "}" )) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "S")) ")" )) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rlaffin2 :::"Int"::: ) (Set (Var "Af"))))) "implies" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Num 2)) ")" & "(" (Bool (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "S")) ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_rlaffin2 :::"Int"::: ) (Set (Var "Af"))))) "implies" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ")" )) ")" )) "holds" (Bool "for" (Set (Var "S")) "being" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Set (Var "n")) ($#k5_real_1 :::"-"::: ) (Num 1)) "," (Set ($#k4_simplex1 :::"BCS"::: ) (Set (Var "K"))) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set (Var "S1")) where S1 "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Var "n")) "," (Set ($#k4_simplex1 :::"BCS"::: ) (Set (Var "K"))) : (Bool (Set (Var "S")) ($#r1_tarski :::"c="::: ) (Set (Var "S1"))) "}" )) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "S")) ")" )) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rlaffin2 :::"Int"::: ) (Set (Var "Af"))))) "implies" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Num 2)) ")" & "(" (Bool (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "S")) ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_rlaffin2 :::"Int"::: ) (Set (Var "Af"))))) "implies" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ")" ))))))) ; theorem :: SIMPLEX1:45 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Aff")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "S")) "being" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Set (Var "n")) ($#k5_real_1 :::"-"::: ) (Num 1)) "," (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "k")) "," (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Aff"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set (Var "S1")) where S1 "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Var "n")) "," (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "k")) "," (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) : (Bool (Set (Var "S")) ($#r1_tarski :::"c="::: ) (Set (Var "S1"))) "}" )) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "S")) ")" )) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rlaffin2 :::"Int"::: ) (Set (Var "Aff"))))) "implies" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Num 2)) ")" & "(" (Bool (Bool (Set ($#k3_convex1 :::"conv"::: ) (Set "(" ($#k1_simplex1 :::"@"::: ) (Set (Var "S")) ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_rlaffin2 :::"Int"::: ) (Set (Var "Aff"))))) "implies" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ")" )))))) ; begin theorem :: SIMPLEX1:46 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Aff")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_simplex0 :::"Vertices"::: ) (Set "(" ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "k")) "," (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ")" ) ")" ) "," (Set (Var "Aff")) "st" (Bool (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set "(" ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "k")) "," (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ")" ) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "Aff"))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "B"))))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) ")" )) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "S")) where S "is" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Aff")) ")" ) ($#k5_real_1 :::"-"::: ) (Num 1)) "," (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "k")) "," (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) : (Bool (Set (Set (Var "F")) ($#k7_relset_1 :::".:"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set (Var "Aff"))) "}" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Num 1)))))))) ; theorem :: SIMPLEX1:47 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "Aff")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_rlaffin1 :::"affinely-independent"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_simplex0 :::"Vertices"::: ) (Set "(" ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "k")) "," (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ")" ) ")" ) "," (Set (Var "Aff")) "st" (Bool (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set "(" ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "k")) "," (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ")" ) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "Aff"))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set ($#k3_convex1 :::"conv"::: ) (Set (Var "B"))))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) ")" )) "holds" (Bool "ex" (Set (Var "S")) "being" ($#m3_simplex0 :::"Simplex"::: ) "of" (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Aff")) ")" ) ($#k5_real_1 :::"-"::: ) (Num 1)) "," (Set ($#k5_simplex1 :::"BCS"::: ) "(" (Set (Var "k")) "," (Set "(" ($#k5_simplex0 :::"Complex_of"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Aff")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) "st" (Bool (Set (Set (Var "F")) ($#k7_relset_1 :::".:"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set (Var "Aff")))))))) ;