reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;
reserve V,W for finite-rank free Z_Module;
reserve T for linear-transformation of V,W;

theorem LMFirst5:
  for R being Ring
  for V being LeftMod of R, W being Subspace of V holds
  ker ZQMorph(V, W) = (Omega).W
  proof
    let R be Ring;
    let V be LeftMod of R, W be Subspace of V;
    set f = ZQMorph(V, W);
    reconsider Ws = (Omega).W as strict Subspace of V by VECTSP_4:26;
    for x being object holds
    x in f" {0.VectQuot(V, W)} iff x in the carrier of W
    proof
      let x be object;
      hereby
        assume A11: x in f" {0.VectQuot(V, W)}; then
        A1: x in the carrier of V & f.x in {0.VectQuot(V, W)}
        by FUNCT_2:38;
        reconsider v = x as Vector of V by A11;
        f.v = 0.VectQuot(V, W) by A1,TARSKI:def 1
        .= zeroCoset(V,W) by VECTSP10:def 6
        .= the carrier of W;
        then v + W = the carrier of W by defMophVW;
        then v in W by VECTSP_4:49;
        hence x in the carrier of W;
      end;
      assume B1: x in the carrier of W;
      B4: the carrier of W c= the carrier of V by VECTSP_4:def 2;
      then reconsider v = x as Vector of V by B1;
      B2: v in W by B1;
      f.v = v + W by defMophVW
      .= zeroCoset(V,W) by B2,VECTSP_4:49
      .= 0.VectQuot(V, W) by VECTSP10:def 6;
      then f.x in {0.VectQuot(V, W)} by TARSKI:def 1;
      hence x in f" {0.VectQuot(V, W)} by B1,B4,FUNCT_2:38;
    end;
    then f" {0.VectQuot(V, W)} = the carrier of W by TARSKI:2;
    then ker f = Ws by LMFirst2,VECTSP_4:29;
    hence thesis;
  end;
