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

theorem Th4:
  for k,m,l be Nat holds k <= l & l <= m implies k = l or k + 1 <= l & l <= m
proof
  defpred P[Nat] means for m,l be Nat holds $1 <= l & l <= m implies $1 = l or
  ($1 + 1 <= l & l <= m);
A1: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
    P[k];
    let m,l be Nat;
    assume that
A2: k+1 <= l and
A3: l <= m;
    k+1 = l or k+1 < l by A2,XXREAL_0:1;
    hence thesis by A3,NAT_1:13;
  end;
A4: P[0] by NAT_1:13;
  thus for k being Nat holds P[k] from NAT_1:sch 2(A4,A1);
end;
