 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;
reserve pnA for Element of(TOP-REAL n)|Affin Affn;

theorem Th28:
  for EN for B be Subset of (TOP-REAL n)|Affin Affn st
    k < card Affn & B = {pnA: (pnA|--EN)|k in Ak}
  holds Ak is closed iff B is closed
proof
  let E be Enumeration of Affn;
  set TRn=TOP-REAL n;
  set A=Affn;
  set AA=Affin A;
  let B be Subset of TRn|AA such that
   A1: k<card A and
   A2: B={v where v is Element of TRn|AA:(v|--E)|k in Ak};
  set B1={v where v is Element of TRn|AA:(v|--E)|k in Ak`};
  A3: B`c=B1
  proof
   let x be object;
   assume A4: x in B`;
   then reconsider v=x as Element of TRn|AA;
   set vE=v|--E;
   len vE=card A by Th16;
   then len(vE|k)=k by A1,FINSEQ_1:59;
   then A5: vE|k in [#]TOP-REAL k by TOPREAL3:46;
   not v in B by A4,XBOOLE_0:def 5;
   then not vE|k in Ak by A2;
   then vE|k in Ak` by A5,XBOOLE_0:def 5;
   hence thesis;
  end;
  A6: A is non empty by A1;
  B1 c=B`
  proof
   let x be object;
   assume x in B1;
   then consider v be Element of TRn|AA such that
    A7: x=v and
    A8: (v|--E)|k in Ak`;
   assume not x in B`;
   then v in B by A6,A7,XBOOLE_0:def 5;
   then ex w be Element of TRn|AA st w=v & (w|--E)|k in Ak by A2;
   hence contradiction by A8,XBOOLE_0:def 5;
   set vE=v|--E;
  end;
  then B1=B` by A3;
  then Ak` is open iff B` is open by A1,Th27;
  hence thesis by TOPS_1:3;
 end;
