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

theorem Th15:
  for D be non empty set holds addpfunc(D) is associative
proof
  let D be non empty set;
  let F1,F2,F3 be Element of PFuncs(D,REAL);
  set o=addpfunc(D);
  thus o.(F1,o.(F2,F3)) = o.(F1,F2+F3) by Def4
    .= F1+(F2+F3) by Def4
    .= F1+F2+F3 by RFUNCT_1:8
    .= o.(F1,F2) + F3 by Def4
    .= o.(o.(F1,F2),F3) by Def4;
end;
