
theorem Th2:
  for n being even Integer, m being odd Integer st m <= n holds m+1 <= n
proof
  let n be even Integer, m be odd Integer;
  assume A1: m <= n;
  consider k1 being Integer such that
    A2: n = 2*k1 by ABIAN:11;
  consider k2 being Integer such that
    A3: m+1 = 2*k2 by ABIAN:11;
  per cases;
  suppose k2 <= k1;
    hence m+1 <= n by A2, A3, XREAL_1:64;
  end;
  suppose k2 > k1;
    then k1 + 1 <= k2 by INT_1:7;
    then A4: 2*(k1+1) <= 2*k2 by XREAL_1:64;
    m+1 <= (2*k1)+1 by A1, A2, XREAL_1:6;
    then 2*k1+2 <= 2*k1+1 by A3, A4, XXREAL_0:2;
    hence thesis by XREAL_1:6;
  end;
end;
