 reserve x for set,
         n,m,k for Nat,
         r for Real,
         V for RealLinearSpace,
         v,u,w,t for VECTOR of V,
         Av for finite Subset of V,
         Affv for finite affinely-independent Subset of V;
reserve pn for Point of TOP-REAL n,
        An for Subset of TOP-REAL n,
        Affn for affinely-independent Subset of TOP-REAL n,
        Ak for Subset of TOP-REAL k;
reserve EV for Enumeration of Affv,
        EN for Enumeration of Affn;

theorem Th26:
  for EN st k <=n & card Affn = n+1 & An = {pn:(pn|--EN)|k in Ak}
    holds Ak is closed iff An is closed
 proof
  set TRn=TOP-REAL n;
  set TRk=TOP-REAL k;
  set A=Affn;
  let E be Enumeration of A such that
   A1: k<=n & card A=n+1 and
   A2: An={v where v is Element of TRn:(v|--E)|k in Ak};
  set B1={v where v is Element of TRn:(v|--E)|k in Ak`};
  A3: k<card A by A1,NAT_1:13;
  A4: An`c=B1
  proof
   let x be object;
   assume A5: x in An`;
   then reconsider f=x as Element of TRn;
   set fE=f|--E;
   len fE=card A by Th16;
   then len(fE|k)=k by A3,FINSEQ_1:59;
   then A6: fE|k is Element of TRk by TOPREAL3:46;
   assume not x in B1;
   then not fE|k in Ak`;
   then fE|k in Ak by A6,XBOOLE_0:def 5;
   then f in An by A2;
   hence contradiction by A5,XBOOLE_0:def 5;
  end;
  B1 c=An`
  proof
   let x be object;
   assume x in B1;
   then consider v be Element of TRn such that
    A7: x=v and
    A8: (v|--E)|k in Ak`;
   assume not x in An`;
   then v in An by A7,XBOOLE_0:def 5;
   then ex w be Element of TRn st v=w & (w|--E)|k in Ak by A2;
   hence contradiction by A8,XBOOLE_0:def 5;
  end;
  then An`=B1 by A4;
  then Ak` is open iff An` is open by A1,Th25;
  hence thesis by TOPS_1:3;
 end;
