theorem VS10Th1:
  for a being Element of INT.Ring, V being add-associative right_zeroed
  right_complementable vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty ModuleStr over INT.Ring,
      v being Vector of V
  holds (0.INT.Ring)*v = 0.V & a*(0.V) = 0.V
  proof
    let x be Element of INT.Ring;
    let V be add-associative right_zeroed right_complementable
    vector-distributive scalar-distributive scalar-associative scalar-unital
    non empty ModuleStr over INT.Ring, v be Vector of V;
    A1: v+(0.INT.Ring)*v = 1.INT.Ring*v + (0.INT.Ring)*v by VECTSP_1:def 17
    .= (1.INT.Ring+0.INT.Ring)*v by VECTSP_1:def 15
    .= v + 0.V by VECTSP_1:def 17;
    hence (0.INT.Ring)*v = 0.V by RLVECT_1:8;
    hence x*(0.V) = (x*0.INT.Ring)*v by VECTSP_1:def 16
    .= 0.V by A1,RLVECT_1:8;
  end;
