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 Th8:
  for x,y being FinSequence of K,a being Element of K st len x=len
  y holds mlt(a*x,y)=a*(mlt(x,y)) & mlt(x,a*y)=a*(mlt(x,y))
proof
  let x,y be FinSequence of K,a be Element of K;
  assume len x=len y;
  then reconsider y0=y as Element of (len x)-tuples_on (the carrier of K) by
Th1;
  reconsider x0=x as Element of (len x)-tuples_on (the carrier of K) by Th1;
  thus mlt(a*x,y)=a*(mlt(x0,y0)) by FVSUM_1:69
    .=a*(mlt(x,y));
  thus mlt(x,a*y)=a*(mlt(x0,y0)) by FVSUM_1:69
    .=a*(mlt(x,y));
end;
