reserve FS for non empty doubleLoopStr;
reserve F for Field;
reserve R for Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr,
  x, y, z for Scalar of R;
reserve SF for Skew-Field,
  x, y, z for Scalar of SF;
reserve R, R1, R2 for Ring;

theorem Th23:
  for x,y being Scalar of R for v,w being Vector of LeftModule R
  holds x*(v+w) = x*v+x*w & (x+y)*v = x*v+y*v & (x*y)*v = x*(y*v) & (1.R)*v = v
proof
  set MLT = the multF of R;
  set LS = ModuleStr (# the carrier of R,the addF of R, 0.R,MLT #);
  for x,y being Scalar of R for v,w being Vector of LS holds x*(v+w) = x*v
  +x*w & (x+y)*v = x*v+y*v & (x*y)*v = x*(y*v) & (1_R)*v = v
  proof
    let x,y be Scalar of R;
    let v,w be Vector of LS;
    reconsider v9 = v, w9 = w as Scalar of R;
    thus x*(v+w) = x*(v9+w9) .= x*v9+x*w9 by VECTSP_1:def 7
      .= x*v+x*w;
    thus (x+y)*v = (x+y)*v9 .= x*v9+y*v9 by VECTSP_1:def 7
      .= x*v+y*v;
    thus (x*y)*v = (x*y)*v9 .= x*(y*v9) by GROUP_1:def 3
      .= x*(y*v);
    thus (1_R)*v = (1_R)*v9 .= v;
  end;
  hence thesis;
end;
