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

theorem Th13:
  for i,j being natural Number holds i < j + 1 iff i <= j
proof
  let i,j be natural Number;
  thus i < j + 1 implies i <= j by Th8;
  assume
A1: i <= j;
A2: now
A3: j + (- j) = 0 & j + 1 + (- j) = 1;
    assume i = j + 1;
    hence contradiction by A1,A3,XREAL_1:6;
  end;
  i <= j + 1 by A1,Th12;
  hence thesis by A2,XXREAL_0:1;
end;
