reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th21:
  for D be non empty set, f1,f2 be FinSequence of PFuncs(D,REAL)
  holds Sum(f1^f2) = Sum f1 + Sum f2
proof
  let D be non empty set, f1,f2 be FinSequence of PFuncs(D,REAL);
  set o = addpfunc(D);
  o is associative by Th15;
  hence Sum(f1^f2)= addpfunc(D).(Sum f1,Sum f2) by Th18,FINSOP_1:5
    .= Sum f1 + Sum f2 by Def4;
end;
