
theorem Th3:
  for F,G,H being sequence of ExtREAL st G is nonnegative & H
  is nonnegative holds (for n being Element of NAT holds F.n = G.n + H.n )
  implies for n being Nat holds (Ser F).n = (Ser G).n + (Ser H).n
proof
  let F,G,H be sequence of ExtREAL;
  assume that
A1: G is nonnegative and
A2: H is nonnegative;
  defpred P[Nat] means (Ser F).$1 = (Ser G).$1 + (Ser H).$1;
  assume
A3: for n being Element of NAT holds F.n = G.n + H.n;
A4: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A5: (Ser F).k = (Ser G).k + (Ser H).k;
    set n = k+1;
A6: 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;
A7: 0. <= H.(k+1) by A2,SUPINF_2:39;
A8: F.(k+1) = G.(k+1) + H.(k+1) & 0. <= G.(k+1) by A1,A3,SUPINF_2:39;
A9: 0. <= (Ser G).k by A1,SUPINF_2:40;
A10: 0. <= (Ser G).(k+1) by A1,SUPINF_2:40;
A11: 0. <= (Ser H).k by A2,SUPINF_2:40;
    0. <= G.n & 0. <= H.n by A1,A2,SUPINF_2:39;
    then 0. + 0. <= G.n + H.n by XXREAL_3:36;
    then 0. <= F.(k+1) by A3;
    then
    (Ser G).k + (Ser H).k + F.(k + 1) = ((Ser G).k + F.(k+1)) + (Ser H).k
    by A9,A11,XXREAL_3:44
      .= (((Ser G).k + G.(k+1)) + H.(k+1)) + (Ser H).k by A9,A7,A8,XXREAL_3:44
      .= (Ser G).(k+1) + (Ser H).(k+1) by A6,A11,A10,A7,XXREAL_3:44;
    hence thesis by A5,SUPINF_2:def 11;
  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 A3,A12;
  thus for k being Nat holds P[k] from NAT_1:sch 2(A13,A4);
end;
