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 Th41:
  Sk is with_non-empty_elements & card Sk+n <= degree K implies
   (Af is Simplex of n+card Sk,BCS K & (center_of_mass V).:Sk c=Af
iff
   ex Tk st Tk misses Sk & Tk\/Sk is c=-linear with_non-empty_elements &
            card Tk=n+1 & Af = (center_of_mass V).:Sk\/(center_of_mass V).:Tk)
 proof
  set B=center_of_mass V;
  set BK=BCS K;
  assume that
   A1: Sk is with_non-empty_elements and
   A2: card Sk+n<=degree K;
  reconsider nc=n+card Sk as ExtReal;
  A3: nc+1-1=nc by XXREAL_3:22;
  A4: [#]K=the carrier of V by SIMPLEX0:def 10;
  then A5: |.K.|c=[#]K;
  then A6: subdivision(B,K)=BK by Def5;
  A7: nc<=degree BK by A2,A5,Th31;
  hereby assume that
    A8: Af is Simplex of n+card Sk,BK and
    A9: B.:Sk c=Af;
   consider T be c=-linear finite simplex-like Subset-Family of K such that
    A10: Af=B.:T by A6,A8,SIMPLEX0:def 20;
   union T is empty or union T in T by SIMPLEX0:9,ZFMISC_1:2;
   then A11: union T is simplex-like by TOPS_2:def 1;
   then @union T is affinely-independent;
   then reconsider UT=union T as finite affinely-independent Subset of V;
   UT=union@T;
   then conv Af c=conv UT by A10,CONVEX1:30,RLAFFIN2:17;
   then reconsider s1=Af as Simplex of BCS Complex_of{UT} by A8,A11,Th40;
   card Af=nc+1 by A7,A8,SIMPLEX0:def 18;
   then A12: s1 is Simplex of n+card Sk,BCS Complex_of{UT} by A3,SIMPLEX0:48;
   set C=Complex_of{UT};
   reconsider cT=card UT as ExtReal;
   |.C.|c=[#]C;
   then A13: degree C=degree BCS C by Th31;
   degree C=cT-1 & card s1<=degree BCS C+1 by SIMPLEX0:24,26;
   then card s1<=card UT by A13,XXREAL_3:22;
   then nc+1<=card UT by A7,A8,SIMPLEX0:def 18;
   then A14: card Sk+n+1<=card UT by XXREAL_3:def 2;
   the_family_of K is subset-closed & UT in the topology of K by A11;
   then A15: bool UT c=the topology of K by SIMPLEX0:1;
   union@Sk c=union T by A1,A5,A8,A9,A10,Th34,ZFMISC_1:77;
   then consider R be finite Subset-Family of V such that
    A16: R misses Sk & R\/Sk is c=-linear with_non-empty_elements & card R=n+1
and
    A17: union R c=UT and
    A18: Af=(center_of_mass V).:Sk\/(center_of_mass V).:R
      by A1,A9,A12,A14,Th35;
   reconsider R as Subset-Family of K by A4;
   R c=bool union R & bool union R c=bool UT by A17,SIMPLEX0:1,ZFMISC_1:82;
   then R c=bool UT;
   then R c=the topology of K by A15;
   then reconsider R as simplex-like finite Subset-Family of K by SIMPLEX0:14;
   take R;
   thus R misses Sk & R\/Sk is c=-linear with_non-empty_elements & card R=n+1 &
Af=B.:Sk\/B.:R by A16,A18;
  end;
  given T be simplex-like finite Subset-Family of K such that
   A19: T misses Sk and
   A20: T\/Sk is c=-linear with_non-empty_elements and
   A21: card T=n+1 and
   A22: Af=B.:Sk\/B.:T;
  set ST=Sk\/T;
  [#]K=[#]BK by A6,SIMPLEX0:def 20;
  then reconsider BST=B.:ST as Subset of BK by SIMPLEX0:def 10;
  A23: ST is simplex-like by TOPS_2:13;
  then reconsider BST as Simplex of BK by A6,A20,SIMPLEX0:def 20;
  card ST=card Sk+card T by A19,CARD_2:40;
  then card BST=card Sk+n+1 by A20,A21,A23,Th33;
  then B.:Sk\/B.:T=B.:ST & card BST=nc+1 by RELAT_1:120,XXREAL_3:def 2;
  hence thesis by A3,A22,SIMPLEX0:48,XBOOLE_1:7;
 end;
