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;
reserve R for domRing;
reserve V for RightMod of R;
reserve v,u for Vector of V;
reserve A,B for Subset of V;
reserve l for Linear_Combination of A;
reserve f,g for Function of the carrier of V, the carrier of R;

theorem Th68:
  x in A implies x in Lin(A)
proof
  deffunc F(Element of V)=0.R;
  assume
A1: x in A;
  then reconsider v = x as Vector of V;
  consider f being Function of the carrier of V, the carrier of R such that
A2: f.v = 1_R and
A3: for u st u <> v holds f.u = F(u) 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 u st not u in T holds f.u = 0.R
  proof
    take T = {v};
    let u;
    assume not u in T;
    then u <> v by TARSKI:def 1;
    hence thesis by A3;
  end;
  then reconsider f as Linear_Combination of V by Def2;
A4: Carrier(f) c= {v}
  proof
    let x be object;
    assume x in Carrier(f);
    then consider u such that
A5: x = u and
A6: f.u <> 0.R;
    u = v by A3,A6;
    hence thesis by A5,TARSKI:def 1;
  end;
  then reconsider f as Linear_Combination of {v} by Def5;
A7: Sum(f) = v * 1_R by A2,Th32
    .= v by VECTSP_2:def 9;
  {v} c= A by A1,ZFMISC_1:31;
  then Carrier(f) c= A by A4;
  then reconsider f as Linear_Combination of A by Def5;
  Sum(f) = v by A7;
  hence thesis by Th67;
end;
