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 Th10:
  for x,y being FinSequence of K,a being Element of K st len x=len
  y holds |(x,a*y)| = a*|(x,y)|
proof
  let x,y be FinSequence of K,a be Element of K;
  assume len x=len y;
  then Sum mlt(x,a*y) =(Sum (a*(mlt(x,y)))) by Th8;
  then Sum mlt(x,a*y) =a*(Sum mlt(x,y)) by FVSUM_1:73;
  then Sum mlt(x,a*y) =a*|(x,y)| by FVSUM_1:def 9;
  hence thesis by FVSUM_1:def 9;
end;
