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;

theorem Th66:
  mlt(i|->a,R) = a*R & mlt(R,i|->a) = a*R
proof
  thus mlt(i|->a,R) = (the multF of K)[;](a,R) by FINSEQOP:20
    .= a*R by FINSEQOP:22;
  hence thesis by FINSEQOP:33;
end;
