 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem HM8:
  for R being Ring
  for Y be LeftMod of R, A be Subset of Y
  holds Lin(A) is strict Subspace of (Omega).Y
  proof
    let R be Ring;
    let Y be LeftMod of R, A be Subset of Y;
    U1: the carrier of Lin(A) c= the carrier of Y
    & 0.Lin(A) = 0.Y
    & the addF of Lin(A) = (the addF of Y) ||the carrier of Lin(A)
    & the lmult of Lin(A) = (the lmult of Y) |
      [:the carrier of R, the carrier of Lin(A):]
    by VECTSP_4:def 2;
    the carrier of Y = the carrier of ((Omega).Y )
    & 0.Y = 0.((Omega).Y)
    & the addF of Y = the addF of ((Omega).Y)
    & the lmult of Y = the lmult of ((Omega).Y);
    hence Lin(A) is strict Subspace of (Omega).Y by U1,VECTSP_4:def 2;
  end;
