reserve i,j,k for Nat;
reserve K for non empty addLoopStr,
  a for Element of K,
  p for FinSequence of the carrier of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for left_zeroed right_zeroed add-associative right_complementable
  non empty addLoopStr,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for non empty addLoopStr,
  a1,a2 for Element of K,
  p1,p2 for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for Abelian right_zeroed add-associative right_complementable non
  empty addLoopStr,
  R,R1,R2,R3 for Element of i-tuples_on the carrier of K;
reserve K for non empty multMagma,
  a,a9,a1,a2 for Element of K,
  p for FinSequence of the carrier of K,
  R for Element of i-tuples_on the carrier of K;

theorem
  for K being associative non empty multMagma, a1,a2 being Element of
K, R being Element of i-tuples_on the carrier of K holds (a1*a2)*R = a1*(a2*R)
proof
  let K be associative non empty multMagma, a1,a2 be Element of K, R be
  Element of i-tuples_on the carrier of K;
  set F=the multF of K;
  set f=id the carrier of K;
  thus (a1*a2)*R = (F[;](a1,F[;](a2,f)))*R by FUNCOP_1:62
    .= ((the multF of K)[;](a1,id the carrier of K)* (the multF of K)[;](a2,
  id the carrier of K))*R by FUNCOP_1:55
    .= a1*(a2*R) by RELAT_1:36;
end;
