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;
reserve K for distributive non empty doubleLoopStr,
  a,a1,a2 for Element of K ,
  R,R1,R2 for Element of i-tuples_on the carrier of K;

theorem
  for K being distributive commutative left_unital non empty
    doubleLoopStr,
      R being Element of i-tuples_on the carrier of K holds
     1.K * R = R
proof
  let K be distributive commutative left_unital non empty doubleLoopStr,
      R be Element of i-tuples_on the carrier of K;
A1: rng R c= the carrier of K by FINSEQ_1:def 4;
  the_unity_wrt the multF of K = 1.K by Th5;
  hence 1.K * R = (id the carrier of K)*R by FINSEQOP:44
    .= R by A1,RELAT_1:53;
end;
