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;

theorem Th8:
  {v} is linearly-independent iff v <> 0.V
proof
  thus {v} is linearly-independent implies v <> 0.V
  proof
    assume {v} is linearly-independent;
    then not 0.V in {v} by Th6;
    hence thesis by TARSKI:def 1;
  end;
  assume
A1: v <> 0.V;
  let l be Linear_Combination of {v};
A2: Carrier(l) c= {v} by RLVECT_2:def 6;
  assume
A3: Sum(l) = 0.V;
  now
    per cases by A2,ZFMISC_1:33;
    suppose
      Carrier(l) = {};
      hence thesis;
    end;
    suppose
A4:   Carrier(l) = {v};
A5:   0.V = l.v * v by A3,RLVECT_2:32;
      now
        assume v in Carrier(l);
        then ex u st v = u & l.u <> 0;
        hence contradiction by A1,A5,RLVECT_1:11;
      end;
      hence thesis by A4,TARSKI:def 1;
    end;
  end;
  hence thesis;
end;
