reserve D,D9 for non empty set;
reserve R for Ring;
reserve G,H,S for non empty ModuleStr over R;
reserve UN for Universe;

theorem Th2:
  for f being Function of G,H, g being Function of H,S st f is
  homogeneous & g is homogeneous holds g*f is homogeneous
proof
  let f be Function of G,H, g be Function of H,S;
  assume that
A1: f is homogeneous and
A2: g is homogeneous;
  set h = g*f;
    let a be Scalar of R, x be Vector of G;
    thus h.(a*x) = g.(f.(a*x)) by FUNCT_2:15
      .= g.(a*f.x) by A1
      .= a*g.(f.x) by A2
      .= a*h.x by FUNCT_2:15;
end;
