reserve

  k,n,m,i,j for Element of NAT,
  K for Field;
reserve L for non empty addLoopStr;
reserve G for non empty multLoopStr;

theorem Th13:
  for p1 being Element of n-tuples_on the carrier of K holds mlt(
  p1,(n |-> (0.K)))=n |-> (0.K)
proof
  let p1 be Element of n-tuples_on (the carrier of K);
  thus mlt(p1,(n |-> (0.K)))= mlt(p1,(0.K)*(0*(K,n))) by FVSUM_1:58
    .= (0.K)*(mlt(p1,0*(K,n))) by FVSUM_1:69
    .= n |-> (0.K) by FVSUM_1:58;
end;
