 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
  for A be affinely-independent Subset of V st |--(A,x) = [#]V-->0
    holds not x in A
 proof
  let A be affinely-independent Subset of V;
  set Ax=|--(A,x);
  assume A1: |--(A,x)=[#]V-->0;
  A2: A c=conv A by RLAFFIN1:2;
  assume A3: x in A;
  then 0=Ax.x by A1,FUNCOP_1:7
   .=(x|--A).x by A3,Def3
   .=1 by A3,A2,RLAFFIN1:72;
  hence contradiction;
 end;
