reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve RR for domRing;
reserve VV for RightMod of RR;
reserve LL for Linear_Combination of VV;
reserve aa for Scalar of RR;
reserve uu, vv for Vector of VV;
reserve R for domRing;
reserve V for RightMod of R;
reserve L,L1,L2 for Linear_Combination of V;
reserve a for Scalar of R;
reserve x for set;
reserve R for Ring;
reserve V for RightMod of R;
reserve v,v1,v2 for Vector of V;
reserve A,B for Subset of V;

theorem Th63:
  0.R <> 1_R & A is linearly-independent implies not 0.V in A
proof
  assume that
A1: 0.R <> 1_R and
A2: A is linearly-independent and
A3: 0.V in A;
  deffunc F(Element of V)=0.R;
  consider f being Function of the carrier of V, the carrier of R such that
A4: f.(0.V) = 1_R and
A5: 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, the carrier of R) by
FUNCT_2:8;
  ex T being finite Subset of V st for v st not v in T holds f.v = 0.R
  proof
    take T = {0.V};
    let v;
    assume not v in T;
    then v <> 0.V by TARSKI:def 1;
    hence thesis by A5;
  end;
  then reconsider f as Linear_Combination of V by Def2;
A6: 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
A7:   v = x and
A8:   f.v <> 0.R;
      v = 0.V by A5,A8;
      hence thesis by A7,TARSKI:def 1;
    end;
    let x be object;
    assume x in {0.V};
    then x = 0.V by TARSKI:def 1;
    hence thesis by A1,A4;
  end;
  then Carrier(f) c= A by A3,ZFMISC_1:31;
  then reconsider f as Linear_Combination of A by Def5;
  Sum(f) = 0.V * f.(0.V) by A6,Th35
    .= 0.V by VECTSP_2:32;
  hence contradiction by A2,A6;
end;
