theorem Th46:
  0.V in A implies
    (A is affinely-independent iff A \ {0.V} is linearly-independent)
 proof
  assume A1: 0.V in A;
  A2: -0.V+A=0.V+A
   .=A by Th6;
  hence A is affinely-independent implies A\{0.V} is linearly-independent by A1
,Th41;
  assume A\{0.V} is linearly-independent;
  hence thesis by A1,A2;
 end;
