reserve D for non empty set,
  i,j,k for Nat,
  n,m for Nat,
  r for Real,
  e for real-valued FinSequence;

theorem Th3:
  (for i be Nat st i in dom e holds 0 <= e.i) implies for f being
Real_Sequence st (for n be Nat st 0 <> n & n < len e holds f.(n+1) = f.n+e.(n+1
  )) holds for n,m be Nat st n in dom e & m in dom e & n <= m holds f.n <= f.m
proof
  assume
A1: for i be Nat st i in dom e holds 0 <= e.i;
  let f being Real_Sequence such that
A2: for n be Nat st 0 <> n & n < len e holds f.(n+1) = f.n+e.(n+1);
A3: for n st n <> 0 & n < len e holds f.n <= f.(n+1)
  proof
    let n such that
A4: n <> 0 and
A5: n < len e;
    n+1 >=1 & n+1 <= len e by A5,NAT_1:13,14;
    then n+1 in dom e by FINSEQ_3:25;
    then f.n + e.(n+1) >= f.n by A1,XREAL_1:31;
    hence thesis by A2,A4,A5;
  end;
  for n be Nat st n in dom e holds for m holds m in dom e & n <=m implies
  f.n <= f.m
  proof
    let n be Nat;
    assume n in dom e;
    then
A6: n >= 1 by FINSEQ_3:25;
    defpred p[Nat] means $1 in dom e & n <= $1 implies f.$1 >= f.n;
A7: now
      let k such that
A8:   p[k];
      now
        assume that
A9:     k + 1 in dom e and
A10:    n <= k + 1;
A11:    k + 1 <= len e by A9,FINSEQ_3:25;
        per cases by A10,A11,NAT_1:8,13;
        suppose
          k + 1 = n & k < len e;
          hence f.(k+1) >= f.n;
        end;
        suppose
A12:      k >= n & k < len e;
          then k >= 1 & f.(k+1) >= f.k by A3,A6,NAT_1:14;
          hence f.(k+1) >= f.n by A8,A12,FINSEQ_3:25,XXREAL_0:2;
        end;
      end;
      hence p[k+1];
    end;
A13: p[0];
    for n be Nat holds p[n] from NAT_1:sch 2(A13,A7);
    hence thesis;
  end;
  hence thesis;
end;
