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 add-associative right_zeroed right_complementable
         distributive non empty doubleLoopStr,
      R being Element of i-tuples_on the carrier of K holds 0.K*R = i|->0.K
proof
  let K be add-associative right_zeroed right_complementable distributive 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;
A2: the addF of K is having_an_inverseOp by Th14;
A3: the_unity_wrt the addF of K = 0.K & the addF of K is having_a_unity by Th7
,Th8;
  thus 0.K*R = (the multF of K)[;](0.K,(id the carrier of K)*R) by FUNCOP_1:34
    .= (the multF of K)[;](0.K,R) by A1,RELAT_1:53
    .= i|->0.K by A3,A2,Th10,FINSEQOP:76;
end;
