theorem Th42:
  A is affinely-independent iff
      for L be Linear_Combination of A st Sum L = 0.V & sum L = 0
        holds Carrier L = {}
 proof
  thus A is affinely-independent implies for L be Linear_Combination of A st
  Sum L=0.V & sum L=0 holds Carrier L={} by Lm5;
  assume for L be Linear_Combination of A st Sum L=0.V & sum L=0 holds Carrier
L={};
  then for p be VECTOR of V st p in A holds(-p+A)\{0.V} is linearly-independent
by Lm6;
  hence thesis by Th41;
 end;
