reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;

theorem Th41:
  for X being Subset of V st X is linearly-dependent holds ex l
  being Linear_Combination of X st Carrier l <> {} & Sum l = 0.V
proof
  let X be Subset of V;
  assume X is linearly-dependent;
  then
  not (for l being Linear_Combination of X st Sum l = 0.V holds Carrier l
  = {}) by VECTSP_7:def 1;
  hence thesis;
end;
