 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 Th32:
  card Affn = n+1 implies |--(Affn,x) is continuous
 proof
  set TRn=TOP-REAL n;
  set AA=Affin Affn;
  set Ax=|--(Affn,x);
  reconsider AxA=Ax|AA as continuous Function of TRn|AA,R^1 by Th31;
  assume A1: card Affn=n+1;
  dim TRn=n by Th4;
  then A2: AA=[#]TRn by A1,Th6;
  then A3: TRn|AA=the TopStruct of TRn by TSEP_1:93;
  A4: dom Ax=AA by A2,FUNCT_2:def 1;
  now let P be Subset of R^1;
   assume P is closed;
   then AxA"P is closed by PRE_TOPC:def 6;
   then A5: (AxA"P)` in the topology of the TopStruct of TRn by A3,
PRE_TOPC:def 2;
   (AxA"P)`=(Ax"P)` by A4,A3,RELAT_1:69;
   then (Ax"P)` is open by A5,PRE_TOPC:def 2;
   hence Ax"P is closed by TOPS_1:3;
  end;
  hence thesis by PRE_TOPC:def 6;
 end;
