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 Th11:
  for x,y1,y2 being FinSequence of K,a being Element of K st len x
  =len y1 & len x=len y2 holds |(x,y1+y2)| = |(x,y1)| + |(x,y2)|
proof
  let x,y1,y2 be FinSequence of K,a be Element of K;
  reconsider x0=x as Element of (len x)-tuples_on (the carrier of K) by Th1;
  assume
A1: len x=len y1 & len x=len y2;
  then reconsider
  y10=y1,y20=y2 as Element of (len x)-tuples_on (the carrier of K)
  by Th1;
  Sum mlt(x,y1+y2) = Sum (mlt(x0,y10) + mlt(x0,y20)) by A1,MATRIX_4:57;
  then Sum mlt(x,y1+y2) = Sum mlt(x0,y10) + Sum mlt(x0,y20) by FVSUM_1:76;
  then Sum mlt(x,y1+y2) = Sum mlt(x,y1) + |(x,y2)| by FVSUM_1:def 9;
  then Sum mlt(x,y1+y2) = |(x,y1)| + |(x,y2)| by FVSUM_1:def 9;
  hence thesis by FVSUM_1:def 9;
end;
