theorem
  for A,B be finite Subset of V st
      A is affinely-independent & Affin A c= Affin B & card A = card B
    holds B is affinely-independent
 proof
  let A,B be finite Subset of V such that
   A1: A is affinely-independent & Affin A c=Affin B & card A=card B;
  {}V c=B;
  then consider Ib be affinely-independent Subset of V such that
   {}V c=Ib and
   A2: Ib c=B and
   A3: Affin Ib=Affin B by Th60;
  reconsider IB=Ib as finite Subset of V by A2;
  A4: card IB<=card B by A3,Th79;
  card B<=card IB by A1,A3,Th79;
  hence thesis by A2,A4,CARD_2:102,XXREAL_0:1;
 end;
