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 LMFirst1:
  for R being Ring
  for V, U, W being LeftMod of R,
  f being linear-transformation of V, U,
  g being linear-transformation of U, W holds
  g*f is linear-transformation of V, W
  proof
    let R be Ring;
    let V, U, W be LeftMod of R,
    f be linear-transformation of V, U,
    g be linear-transformation of U, W;
    set gf = g*f;
    for a being Element of R,
        x being Element of V holds gf.(a*x) = a*(gf.x)
    proof
      let a be Element of R, x be Element of V;
      P3: f is homogeneous;
      P4: g is homogeneous;
      thus gf.(a*x) = g.(f.(a*x)) by FUNCT_2:15
      .= g.(a*f.x) by P3
      .= a*g.(f.x) by P4
      .= a*gf.x by FUNCT_2:15;
    end;
    then gf is homogeneous;
    hence thesis;
  end;
