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 Th71:
  for W being strict Submodule of V st 0.R <> 1_R & A = the
  carrier of W holds Lin(A) = W
proof
  let W be strict Submodule of V;
  assume that
A1: 0.R <> 1_R and
A2: A = the carrier of W;
  now
    let v;
    thus v in Lin(A) implies v in W
    proof
      assume v in Lin(A);
      then
A3:   ex l st v = Sum(l) by Th67;
      A is linearly-closed by A2,RMOD_2:33;
      then v in the carrier of W by A1,A2,A3,Th29;
      hence thesis by STRUCT_0:def 5;
    end;
    v in W iff v in the carrier of W by STRUCT_0:def 5;
    hence v in W implies v in Lin(A) by A2,Th68;
  end;
  hence thesis by RMOD_2:30;
end;
