reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;

theorem Th15:
  for x1,x2,y1,y2 being FinSequence of REAL 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 REAL;
A1: Sum ((mlt(x1,x2))^(mlt(y1,y2)))=Sum mlt(x1,x2) + Sum mlt(y1,y2) by
RVSUM_1:75;
  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,Th14;
  then |( x1^y1, x2^y2 )| = Sum mlt(x1,x2) + Sum mlt(y1,y2) by RVSUM_1:def 16;
  then |( x1^y1, x2^y2 )| = Sum mlt(x1,x2) + |(y1,y2)| by RVSUM_1:def 16;
  hence thesis by RVSUM_1:def 16;
end;
