
theorem Th4:
  for F,G,H being sequence of ExtREAL st for n being Element of
NAT holds F.n = G.n + H.n holds G is nonnegative & H is nonnegative implies SUM
  (F) <= SUM(G) + SUM(H)
proof
  let F,G,H be sequence of ExtREAL;
  assume
A1: for n being Element of NAT holds F.n = G.n + H.n;
  defpred P[Nat] means Ser(F).$1 = Ser(G).$1 + Ser(H).$1;
  assume that
A2: G is nonnegative and
A3: H is nonnegative;
A4: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
A5: Ser(G).(k+1) = Ser(G).k + G.(k+1) & Ser(H).(k+1) = Ser(H).k + H.(k+1)
    by SUPINF_2:def 11;
A6: 0. <= Ser(G).k by A2,SUPINF_2:40;
A7: 0. <= Ser(H).(k+1) by A3,SUPINF_2:40;
A8: 0. <= H.(k+1) by A3,SUPINF_2:39;
A9: 0. <= Ser(H).k by A3,SUPINF_2:40;
    0. <= G.(k+1) & 0. <= H.(k+1) by A2,A3,SUPINF_2:39;
    then 0. + 0. <= G.(k+1) + H.(k+1) by XXREAL_3:36;
    then
A10: Ser(F).(k+1) = Ser(F).k + F.(k+1) & 0. <= F.(k+1) by A1,SUPINF_2:def 11;
A11: 0. <= G.(k+1) by A2,SUPINF_2:39;
    assume Ser(F).k = Ser(G).k + Ser(H).k;
    hence Ser(F).(k+1) = Ser(G).k + (Ser(H).k + F.(k+1)) by A10,A6,A9,
XXREAL_3:44
      .= Ser(G).k + (Ser(H).k + (G.(k+1) + H.(k+1))) by A1
      .= Ser(G).k + ((Ser(H).k + H.(k+1)) + G.(k+1)) by A11,A9,A8,XXREAL_3:44
      .= Ser(G).(k+1) + Ser(H).(k+1) by A5,A6,A11,A7,XXREAL_3:44;
  end;
A12: Ser(H).0 = H.0 by SUPINF_2:def 11;
  Ser(F).0 = F.0 & Ser(G).0 = G.0 by SUPINF_2:def 11;
  then
A13: P[0] by A1,A12;
A14: for n being Nat holds P[n] from NAT_1:sch 2(A13,A4);
  SUM(G) + SUM(H) is UpperBound of rng Ser(F)
  proof
    let x be ExtReal;
A15: dom Ser(F) = NAT by FUNCT_2:def 1;
    assume x in rng Ser(F);
    then consider n being object such that
A16: n in NAT and
A17: x = Ser(F).n by A15,FUNCT_1:def 3;
    reconsider n as Element of NAT by A16;
    Ser(H).n <= sup(rng Ser(H)) by FUNCT_2:4,XXREAL_2:4;
    then
A18: Ser(H).n <= SUM(H);
    Ser(G).n <= sup(rng Ser(G)) by FUNCT_2:4,XXREAL_2:4;
    then Ser(G).n <= SUM(G);
    then Ser(G).n + Ser(H).n <= SUM(G) + SUM(H) by A18,XXREAL_3:36;
    hence thesis by A14,A17;
  end;
  then sup(rng Ser(F)) <= SUM(G) + SUM(H) by XXREAL_2:def 3;
  hence thesis;
end;
