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;
reserve Ks for simplex-join-closed SimplicialComplex of V,
        As,Bs for Subset of Ks,
        Ka for non void affinely-independent SimplicialComplex of V,
        Kas for non void affinely-independent simplex-join-closed
                 SimplicialComplex of V,
        K for non void affinely-independent simplex-join-closed total
                 SimplicialComplex of V;
reserve Aff for finite affinely-independent Subset of V,
        Af,Bf for finite Subset of V,
        B for Subset of V,
        S,T for finite Subset-Family of V,
        Sf for c=-linear finite finite-membered Subset-Family of V,
        Sk,Tk for finite simplex-like Subset-Family of K,
        Ak for Simplex of K;

theorem Th37:
  for S st S is c=-linear & S is with_non-empty_elements &
           card S = card union S
  for Af,Bf st Af is non empty & Af misses union S &
               union S\/Af is affinely-independent & union S\/Af c=Bf
   holds (center_of_mass V).:S \/ (center_of_mass V).:{union S\/Af}
           is Simplex of card S,BCS Complex_of{Bf}
 proof
  let S be finite Subset-Family of V such that
   A1: S is c=-linear & S is with_non-empty_elements and
   A2: card S=card union S;
  set U=union S,b=center_of_mass V;
  let A,B be finite Subset of V such that
   A3: A is non empty and
   A4: A misses U & U\/A is affinely-independent and
   A5: U\/A c=B;
  reconsider UA=U\/A as finite Subset of V by A5;
  dom b=(bool the carrier of V)\{{}} by FUNCT_2:def 1;
  then UA in dom b by A3,ZFMISC_1:56;
  then A6: {b.UA}=Im(b,UA) by FUNCT_1:59
   .=b.:{UA} by RELAT_1:def 16;
  set CA=Complex_of{UA};
  set CB=Complex_of{B};
  {UA}is_finer_than{B}
  proof
   let x;
   assume x in {UA};
   then A7: x=UA by TARSKI:def 1;
   B in {B} by TARSKI:def 1;
   hence thesis by A5,A7;
  end;
  then CA is SubSimplicialComplex of CB by SIMPLEX0:30;
  then A8: subdivision(b,CA) is SubSimplicialComplex of subdivision(b,CB) by
SIMPLEX0:58;
  |.CA.|c=[#]CA;
  then A9: subdivision(b,CA)=BCS CA by Def5;
  |.CB.|c=[#]CB;
  then A10: BCS CA is SubSimplicialComplex of BCS CB by A8,A9,Def5;
  S is finite-membered
  proof
   let x;
   assume x in S;
   then A11: x c=union S by ZFMISC_1:74;
   union S is finite by A2;
   hence thesis by A11;
  end;
  then b.:S\/b.:{UA} is Simplex of card S,BCS CA by A1,A2,A3,A4,Lm1;
  hence thesis by A6,A10,SIMPLEX0:49;
 end;
