 reserve x,y for object, X,Y,Z for set;
 reserve GF for commutative
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr;
 reserve a,b for Element of GF;
 reserve V for scalar-distributive vector-distributive
   scalar-associative scalar-unital add-associative right_zeroed
     right_complementable Abelian non empty ModuleStr over GF;
 reserve v,v1,v2,u for Vector of V;
 reserve A,B,C for Subset of V;
 reserve T for finite Subset of V;
 reserve l for Linear_Combination of A;
 reserve f,g for Function of V, GF;

theorem
  A c= B & B is linearly-independent implies A is linearly-independent
proof
  assume that
A1: A c= B and
A2: B is linearly-independent;
  let l be Linear_Combination of A;
  reconsider L = l as Linear_Combination of B by A1,VECTSP_6:4;
  assume Sum(l) = 0.V;
  then Carrier(L) = {} by A2;
  hence thesis;
end;
