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

theorem Th20:
  for D be non empty set, f be FinSequence of PFuncs(D,REAL), G be
  Element of PFuncs(D,REAL) holds Sum(f^<*G*>) = Sum f + G
proof
  let D be non empty set, f be FinSequence of PFuncs(D,REAL), G be Element of
  PFuncs(D,REAL);
  set o = addpfunc(D);
  thus Sum(f^<*G*>) = o.(o $$ f,G) by Th18,FINSOP_1:4
    .= Sum f + G by Def4;
end;
