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);

theorem
  Lin(A) = (0).V implies A = {} or A = {0.V}
proof
  assume that
A1: Lin(A) = (0).V and
A2: A <> {};
  thus A c= {0.V}
  proof
    let x be object;
    assume x in A;
    then x in Lin(A) by Th15;
    then x = 0.V by A1,Lm2;
    hence thesis by TARSKI:def 1;
  end;
  set y = the Element of A;
  let x be object;
  assume x in {0.V};
  then
A3: x = 0.V by TARSKI:def 1;
  y in A & y in Lin(A) by A2,Th15;
  hence thesis by A1,A3,Lm2;
end;
