theorem Th23:
  (a + b) * g = a * g + (b * g)
proof
  thus (a + b) * g = (-g) + (a + 0_G + b) + g by Def4
    .= (-g) + (a + (g + (-g)) + b) + g by Def5
    .= (-g) + (a + g + (-g) + b) + g by RLVECT_1:def 3
    .= (-g) + (a + g + ((-g) + b)) + g by RLVECT_1:def 3
    .= (-g) + (a + g) + ((-g) + b) + g by RLVECT_1:def 3
    .= a * g + ((-g) + b) + g by RLVECT_1:def 3
    .= a * g + (b * g) by RLVECT_1:def 3;
end;
