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;
