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;
