theorem Th163:
  for R being Abelian right_zeroed add-associative right_complementable
    non empty addLoopStr,
  a being Element of R, i, j being Element of INT.Ring holds
  (Int-mult-left(R)).(i*j,a) = (Int-mult-left(R)).(i,(Int-mult-left(R)).(j,a))
  proof
    let R be Abelian right_zeroed add-associative right_complementable
    non empty addLoopStr,
    a be Element of R, i, j be Element of INT.Ring;
    per cases;
    suppose i = 0 or j = 0;
      hence (Int-mult-left(R)).(i*j,a)
      =(Int-mult-left(R)).(i,(Int-mult-left(R)).(j,a)) by Lm20;
    end;
    suppose not (i = 0 or j = 0);
      hence (Int-mult-left(R)).(i*j,a)
      =(Int-mult-left(R)).(i,(Int-mult-left(R)).(j,a)) by Lm19;
    end;
  end;
