reserve x,y for object, X,Y,Z for set;
reserve a,b for Real;
reserve k for Element of NAT;
reserve V for RealLinearSpace;
reserve W1,W2,W3 for Subspace of V;
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,L1,L2 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve F,G,H for FinSequence of the carrier of V;
reserve f,g for Function of the carrier of V, REAL;
reserve p,q,r for FinSequence;
reserve M for non empty set;
reserve CF for Choice_Function of M;
reserve l0 for Linear_Combination of {}(the carrier of V);
reserve I for Basis of V;

theorem
  for V being strict RealLinearSpace,A being Subset of V holds A is
  linearly-independent implies ex I being Basis of V st A c= I
proof
  let V be strict RealLinearSpace,A be Subset of V;
  assume A is linearly-independent;
  then consider B being Subset of V such that
A1: A c= B and
A2: B is linearly-independent & Lin(B) = V by Th24;
  reconsider B as Basis of V by A2,Def3;
  take B;
  thus thesis by A1;
end;
