theorem Th41:
  A is affinely-independent iff for v st v in A holds
                                      -v + A\{0.V} is linearly-independent
 proof
  hereby assume A is affinely-independent;
   then for L be Linear_Combination of A st Sum L=0.V & sum L=0 holds Carrier L
={} by Lm5;
   hence for v st v in A holds(-v+A)\{0.V} is linearly-independent by Lm6;
  end;
  assume A1: for v st v in A holds(-v+A)\{0.V} is linearly-independent;
  assume A is non empty;
  then consider v be object such that
   A2: v in A;
  reconsider v as Element of V by A2;
  take v;
  thus thesis by A1,A2;
 end;
