
theorem Th2:
  for F being add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr, x being Element of F holds
  x*(0.F) = 0.F
proof
  let F be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr;
  let x be Element of F;
  x*(0.F)+(0.F) = x*((0.F)+(0.F))+(0.F) by RLVECT_1:4
    .= x*((0.F)+(0.F)) by RLVECT_1:4
    .= x*(0.F)+x*(0.F) by Def2;
  hence thesis by RLVECT_1:8;
end;
