
theorem Th1V:
  for R being commutative Ring
  for V being LeftMod of R, W being Subspace of V
  holds (1.R) (*) W = (Omega).W
  proof
    let R be commutative Ring;
    let V be LeftMod of R, W be Subspace of V;
    for v being Vector of V st v in (Omega).W holds v in (1.R) (*) W
    proof
      let v be Vector of V such that
      B1: v in (Omega).W;
      reconsider vv = v as Vector of W by B1;
      (1.R) * vv in (1.R) (*) W;
      hence thesis;
    end; then
    A2: for v being Vector of V holds v in (1.R) (*) W iff
      v in (Omega).W;
    A3: (Omega).W is Subspace of V by ZMODUL01:42;
    (1.R) (*) W is Subspace of V by ZMODUL01:42;
    hence thesis by A2,A3,ZMODUL01:46;
  end;
