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 Th6:
  A is linearly-independent implies not 0.V in A
proof
  assume that
A1: A is linearly-independent and
A2: 0.V in A;
  deffunc F(Element of V) = In(0,REAL);
  consider f such that
A3: f.(0.V) = jj and
A4: for v being Element of V st v <> 0.V holds f.v = F(v) from FUNCT_2:
  sch 6;
  reconsider f as Element of Funcs(the carrier of V, REAL) by FUNCT_2:8;
  ex T st for v st not v in T holds f.v = 0
  proof
    take T = {0.V};
    let v;
    assume not v in T;
    then v <> 0.V by TARSKI:def 1;
    hence thesis by A4;
  end;
  then reconsider f as Linear_Combination of V by RLVECT_2:def 3;
A5: Carrier(f) = {0.V}
  proof
    thus Carrier(f) c= {0.V}
    proof
      let x be object;
      assume x in Carrier(f);
      then consider v such that
A6:   v = x and
A7:   f.v <> 0;
      v = 0.V by A4,A7;
      hence thesis by A6,TARSKI:def 1;
    end;
    let x be object;
    assume x in {0.V};
    then x = 0.V by TARSKI:def 1;
    hence thesis by A3;
  end;
  then Carrier(f) c= A by A2,ZFMISC_1:31;
  then reconsider f as Linear_Combination of A by RLVECT_2:def 6;
  Sum(f) = f.(0.V) * 0.V by A5,RLVECT_2:35
    .= 0.V;
  hence contradiction by A1,A5;
end;
