let V be RealLinearSpace; :: thesis: for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds

BCS (0,Kv) = Kv

let Kv be non void SimplicialComplex of V; :: thesis: ( |.Kv.| c= [#] Kv implies BCS (0,Kv) = Kv )

assume |.Kv.| c= [#] Kv ; :: thesis: BCS (0,Kv) = Kv

hence BCS (0,Kv) = subdivision (0,(center_of_mass V),Kv) by Def6

.= Kv by SIMPLEX0:61 ;

:: thesis: verum

BCS (0,Kv) = Kv

let Kv be non void SimplicialComplex of V; :: thesis: ( |.Kv.| c= [#] Kv implies BCS (0,Kv) = Kv )

assume |.Kv.| c= [#] Kv ; :: thesis: BCS (0,Kv) = Kv

hence BCS (0,Kv) = subdivision (0,(center_of_mass V),Kv) by Def6

.= Kv by SIMPLEX0:61 ;

:: thesis: verum