reserve n for non zero Nat,
  j,k,l,m for Nat,
  g,h,i for Integer;

theorem
  for m be Nat holds 0*m + 0*m = 0*m
proof
  let m be Nat;
A1: dom(0*m) = Seg m by FUNCOP_1:13;
  dom addreal = [:REAL,REAL:] & rng 0*m c= REAL by FINSEQ_1:def 4,FUNCT_2:def 1
;
  then
A2: [:rng 0*m, rng 0*m:] c= dom addreal by ZFMISC_1:96;
A3: dom(0*m + 0*m) = dom(addreal.:(0*m,0*m)) by RVSUM_1:def 4
    .= dom 0*m /\ dom 0*m by A2,FUNCOP_1:69
    .= Seg m by FUNCOP_1:13;
  for k be Nat st k in dom(0*m) holds (0*m).k = (0*m+0*m).k
  proof
    let k be Nat such that
A4: k in dom(0*m);
    (0*m).k = 0;
    then (0*m+0*m).k = 0 + 0 by A3,A1,A4,VALUED_1:def 1;
    hence thesis;
  end;
  hence thesis by A3,FINSEQ_1:13,FUNCOP_1:13;
end;
