reserve X for set,
  F for Field_Subset of X,
  M for Measure of F,
  A,B for Subset of X,
  Sets for SetSequence of X,
  seq,seq1,seq2 for ExtREAL_sequence,
  n,k for Nat;
reserve FSets for Set_Sequence of F,
  CA for Covering of A,F;
reserve Cvr for Covering of Sets,F;
reserve C for C_Measure of X;

theorem Th23:
  for F,G being sequence of ExtREAL, n being Nat st (for m
  being Nat st m <= n holds F.m <= G.m) holds (Ser F).n <= (Ser G).n
proof
  let F,G be sequence of ExtREAL;
  let n be Nat;
  assume
A1: for m being Nat st m <= n holds F.m <= G.m;
  defpred P[Nat] means (for m being Nat st m <= $1 holds F.m <= G.m) implies (
  Ser F).$1 <= (Ser G).$1;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A3: P[k];
    now
      assume
A4:   for m being Nat st m <= k+1 holds F.m <= G.m;
A5:   now
        let m be Nat;
        assume m <= k;
        then m < k+1 by NAT_1:13;
        hence F.m <= G.m by A4;
      end;
      F.(k+1) <= G.(k+1) by A4;
      then (Ser F).k + F.(k+1) <= (Ser G).k + G.(k+1) by A3,A5,XXREAL_3:36;
      then (Ser F).(k+1) <= (Ser G).k + G.(k+1) by SUPINF_2:def 11;
      hence (Ser F).(k+1) <= (Ser G).(k+1) by SUPINF_2:def 11;
    end;
    hence P[k+1];
  end;
  now
A6: (Ser F).0 = F.0 & (Ser G).0 = G.0 by SUPINF_2:def 11;
    assume for m being Nat st m <= 0 holds F.m <= G.m;
    hence (Ser F).0 <= (Ser G).0 by A6;
  end;
  then
A7: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A7,A2);
  hence thesis by A1;
end;
