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 Th12:
  for x1,x2,y1,y2 being FinSequence of K st len x1=len x2 & len y1
  =len y2 holds |( x1^y1, x2^y2 )| = |(x1,x2)| + |(y1,y2)|
proof
  let x1,x2,y1,y2 be FinSequence of K;
A1: Sum ((mlt(x1,x2))^(mlt(y1,y2)))=Sum mlt(x1,x2) + Sum mlt(y1,y2) by
RLVECT_1:41;
  assume len x1=len x2 & len y1=len y2;
  then Sum mlt(x1^y1,x2^y2) = Sum mlt(x1,x2) + Sum mlt(y1,y2) by A1,Th7;
  then |( x1^y1, x2^y2 )| = Sum mlt(x1,x2) + Sum mlt(y1,y2) by FVSUM_1:def 9;
  then |( x1^y1, x2^y2 )| = Sum mlt(x1,x2) + |(y1,y2)| by FVSUM_1:def 9;
  hence thesis by FVSUM_1:def 9;
end;
