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 Th34:
  for S1,S2 be simplex-like Subset-Family of Kas st
      |.Kas.|c=[#]Kas & S1 is with_non-empty_elements &
      (center_of_mass V).:S2 is Simplex of BCS Kas &
      (center_of_mass V).:S1 c= (center_of_mass V).:S2
   holds S1 c= S2 & S2 is c=-linear
 proof
  set B=center_of_mass V;
  set BK=BCS Kas;
  let S1,S2 be simplex-like Subset-Family of Kas;
  assume that
   A1: |.Kas.|c=[#]Kas and
   A2: S1 is with_non-empty_elements and
   A3: B.:S2 is Simplex of BCS Kas and
   A4: B.:S1 c=B.:S2;
  BK=subdivision(B,Kas) by A1,Def5;
  then consider W2 be c=-linear finite simplex-like Subset-Family of Kas such
that
   A5: B.:S2=B.:W2 by A3,SIMPLEX0:def 20;
  reconsider s2=S2\{{}} as simplex-like Subset-Family of Kas by TOPS_2:11
,XBOOLE_1:36;
  set TK=the topology of Kas;
  set BTK=B|TK;
  A6: dom BTK=dom B/\TK by RELAT_1:61;
  A7: s2 c=TK by SIMPLEX0:14;
  A8: dom B=(bool the carrier of V)\{{}} by FUNCT_2:def 1;
  then @S2\{{}}c=dom B by XBOOLE_1:33;
  then s2 c=dom B/\TK by A7,XBOOLE_1:19;
  then A9: s2 c=dom BTK by RELAT_1:61;
  W2/\dom B c=W2 by XBOOLE_1:17;
  then reconsider w2=W2/\dom B as c=-linear finite simplex-like Subset-Family
of Kas by TOPS_2:11,XBOOLE_1:1;
  A10: w2 c=TK by SIMPLEX0:14;
  then A11: B.:W2=B.:(W2/\dom B) & B.:w2=BTK.:w2 by RELAT_1:112,129;
  W2/\dom B c=dom B by XBOOLE_1:17;
  then A12: w2 c=dom BTK by A6,A10,XBOOLE_1:19;
  S2 c=TK by SIMPLEX0:14;
  then B.:S2=BTK.:S2 by RELAT_1:129;
  then A13: w2 c=S2 by A5,A11,A12,FUNCT_1:87;
  A14: S1 c=TK by SIMPLEX0:14;
  S2/\dom B=(@S2/\bool the carrier of V)\{{}} by A8,XBOOLE_1:49
   .=s2 by XBOOLE_1:28;
  then A15: B.:S2=B.:s2 by RELAT_1:112;
  then BTK.:s2=B.:S2 by A7,RELAT_1:129;
  then A16: s2 c=w2 by A5,A11,A9,FUNCT_1:87;
  @S1 c=bool the carrier of V & not{} in S1 by A2;
  then S1 c=dom B by A8,ZFMISC_1:34;
  then A17: S1 c=dom BTK by A6,A14,XBOOLE_1:19;
  B.:S1=BTK.:S1 by A14,RELAT_1:129;
  then S1 c=w2 by A4,A5,A11,A17,FUNCT_1:87;
  hence S1 c=S2 by A13;
  let x,y such that
   A18: x in S2 & y in S2;
  B.:s2=BTK.:s2 by A7,RELAT_1:129;
  then w2 c=s2 by A5,A11,A12,A15,FUNCT_1:87;
  then A19: s2=w2 by A16;
  per cases;
  suppose x is empty or y is empty;
   then x c=y or y c=x;
   hence thesis;
  end;
  suppose x is non empty & y is non empty;
   then x in w2 & y in w2 by A18,A19,ZFMISC_1:56;
   hence thesis by ORDINAL1:def 8;
  end;
 end;
