 reserve x for Real,
    p,k,l,m,n,s,h,i,j,k1,t,t1 for Nat,
    X for Subset of REAL;

theorem Th8:
  for i,j being natural Number holds i <= j + 1 implies i <= j or i = j + 1
proof
  let i,j be natural Number;
A0: i is Nat & j is Nat by TARSKI:1;
  defpred P[natural Number] means
  for j holds $1 <= j + 1 implies $1 <= j or $1 = j+1;
A1: for i st P[i] holds P[i+1]
  proof
    let i such that
A2: for j holds i <= j + 1 implies i <= j or i = j + 1;
    let j;
    assume
A3: i + 1 <= j + 1;
A4: now
      given k such that
A5:   j = k + 1;
      i <= k + 1 by A3,A5,XREAL_1:6;
      hence thesis by A2,A5,XREAL_1:6;
    end;
    now
A6:   0 <= i by Th2;
      assume
A7:   j = 0;
      then i <= 0 by A3,XREAL_1:6;
      hence thesis by A7,A6;
    end;
    hence thesis by A4,Th6;
  end;
A8: P[0] by Th2;
  for i holds P[i] from NatInd(A8,A1);
  hence thesis by A0;
end;
