theorem Th156:
  for R being Abelian right_zeroed add-associative right_complementable
    non empty addLoopStr,
  a being Element of R, i, j, k being Element of NAT
  st i <= j & k = j-i holds
  (Nat-mult-left(R)).(k,a)
  = (Nat-mult-left(R)).(j,a) - (Nat-mult-left(R)).(i,a)
  proof
    let R be Abelian right_zeroed add-associative
    right_complementable non empty addLoopStr,
    a being Element of R, i, j, k being Element of NAT;
    assume A1: i <= j & k = j-i;
    A2:j*a = (k+i)*a by A1
   .= k*a + i*a by BINOM:15;
   thus (Nat-mult-left(R)).(j,a)-(Nat-mult-left(R)).(i,a)
   = (k*a) +((i*a) -(i*a)) by A2,RLVECT_1:28
   .= (k*a) + 0.R by RLVECT_1:15
   .= (Nat-mult-left(R)).(k,a) by RLVECT_1:4;
 end;
