
theorem Th1:
  for K be add-associative right_zeroed right_complementable
associative left_unital distributive non empty doubleLoopStr, a be Element of
  K for V be add-associative right_zeroed right_complementable
  vector-distributive
  scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over K, v be Vector of V holds (0.K)*v = 0.V & a*(0.V) = 0.V
proof
  let F be add-associative right_zeroed right_complementable associative
  left_unital distributive non empty doubleLoopStr;
  let x be Element of F;
  let V be add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over F, v be Vector of V;
  v+(0.F)*v = (1.F)*v + (0.F)*v by VECTSP_1:def 17
    .= ((1.F)+(0.F))*v by VECTSP_1:def 15
    .= (1.F)*v by RLVECT_1:4
    .= v by VECTSP_1:def 17
    .= v+0.V by RLVECT_1:4;
  hence
A1: (0.F)*v = 0.V by RLVECT_1:8;
  hence x*(0.V) = (x*(0.F))*v by VECTSP_1:def 16
    .= 0.V by A1;
end;
