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 Th45:
  for S be Simplex of n-1,BCS(k,Complex_of{Aff}) st card Aff = n+1 &
      X = {S1 where S1 is Simplex of n,BCS(k,Complex_of{Aff}): S c= S1}
    holds (conv @S meets  Int Aff implies card X = 2) &
          (conv @S misses Int Aff implies card X = 1)
  proof
  let S be Simplex of n-1,BCS(k,Complex_of{Aff}) such that
   A1: card Aff=n+1;
  set C=Complex_of{Aff};
  reconsider cA=card Aff as ExtReal;
  A2: cA-1=card Aff+-1 by XXREAL_3:def 2
   .=n by A1;
  then A3: degree C=n by SIMPLEX0:26;
  defpred P[Nat] means
   for S be Simplex of n-1,BCS($1,C),X be set st
   X={S1 where S1 is Simplex of n,BCS($1,C):S c=S1} holds
   (conv@S meets Int Aff implies card X=2) &
   (conv@S misses Int Aff implies card X=1);
  A4: [#]C=[#]V & |.C.|c=[#]V;
  A5: P[0 qua Nat]
  proof
   reconsider n1=n-1 as ExtReal;
   A6: the topology of C=bool Aff by SIMPLEX0:4;
   Aff in bool Aff by ZFMISC_1:def 1;
   then reconsider A1=Aff as finite Simplex of C by A6,PRE_TOPC:def 2;
   A7: BCS(0,C)=C by A4,Th16;
   let S be Simplex of n-1,BCS(0,C),X be set such that
    A8: X={S1 where S1 is Simplex of n,BCS(0,C):S c=S1};
   A9: X c={Aff}
   proof
    let x be object;
    reconsider N=n as ExtReal;
    assume x in X;
    then consider U be Simplex of n,C such that
     A10: x=U and
     S c=U by A7,A8;
    A11: U in the topology of C by PRE_TOPC:def 2;
    card U=N+1 by A3,SIMPLEX0:def 18
     .=n+1 by XXREAL_3:def 2;
    then Aff=U by A1,A6,A11,CARD_2:102;
    hence thesis by A10,TARSKI:def 1;
   end;
   A12: S in bool Aff by A6,A7,PRE_TOPC:def 2;
   A1 is Simplex of n,C by A2,SIMPLEX0:48;
   then Aff in X by A7,A8,A12;
   then A13: X={Aff} by A9,ZFMISC_1:33;
   n+-1>=-1 & n-1<=degree C by A3,XREAL_1:31,146;
   then card S=n1+1 by A7,SIMPLEX0:def 18
    .=n-1+1 by XXREAL_3:def 2;
   then S<>Aff by A1;
   then S c<Aff by A12;
   hence thesis by A13,CARD_1:30,RLAFFIN2:7;
  end;
  A14: for k st P[k] holds P[k+1]
  proof
   let k such that
    A15: P[k];
   A16: degree BCS(k,C)=n & BCS(k+1,C)=BCS BCS(k,C) by A3,A4,Th20,Th32;
   let S be Simplex of n-1,BCS(k+1,C),X such that
    A17: X={S1 where S1 is Simplex of n,BCS(k+1,C):S c=S1};
   thus thesis by A1,A15,A16,A17,Th44;
  end;
  for k holds P[k] from NAT_1:sch 2(A5,A14);
  hence thesis;
 end;
