reserve x,y,X for set,
        r for Real,
        n,k for Nat;
reserve RLS for non empty RLSStruct,
        Kr,K1r,K2r for SimplicialComplexStr of RLS,
        V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;

theorem Th8:
  for A be Subset of RLS holds |.Complex_of{A}.| = conv A
 proof
  let A be Subset of RLS;
  set C=Complex_of{A};
  reconsider A1=A as Subset of C;
  A1: the topology of C=bool A by SIMPLEX0:4;
  hereby let x be object;
   assume x in |.C.|;
   then consider S be Subset of C such that
    A2: S is simplex-like and
    A3: x in conv@S by Def3;
   S in the topology of C by A2;
   then conv@S c=conv A by A1,RLAFFIN1:3;
   hence x in conv A by A3;
  end;
  A c=A;
  then @A1=A & A1 is simplex-like by A1;
  hence thesis by Th5;
 end;
