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;
reserve K for non empty multMagma,
  a1,a2,b1,b2 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 commutative non empty multMagma,
  p,q for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for commutative associative non empty multMagma,
  a,a1,a2 for Element of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for commutative associative non empty multMagma,
  a for Element of K,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for add-associative right_zeroed right_complementable non empty
  addLoopStr,
  a for Element of K,
  p for FinSequence of the carrier of K;
reserve K for Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr,
  p for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for commutative associative well-unital non empty doubleLoopStr,
  a ,a1,a2,a3 for Element of K,
  p1 for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;

theorem
  for K being right_complementable add-associative right_zeroed
    non empty doubleLoopStr
  for a,b being Element of K holds <*a*> "*" <*b*>= a * b
proof
  let K be right_complementable add-associative right_zeroed
    non empty doubleLoopStr;
  let a,b be Element of K;
  set p = <*a*>, q = <*b*>;
  set m = mlt(p,q);
  m = <* (a * b) *> by FINSEQ_2:74;
  hence thesis by RLVECT_1:44;
end;
