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

theorem Th19:
  for D be non empty set holds Sum(<*> PFuncs(D,REAL) ) = [#](D)
  -->(0 qua Real)
proof
  let D be non empty set;
  set o = addpfunc(D), o0 = [#](D) --> (0 qua Real);
  the_unity_wrt o = o0 by Th17;
  hence thesis by Th18,FINSOP_1:10;
end;
