
theorem Th1:
  for n being even Integer, m being odd Integer st n <= m holds n+1 <= m
proof
  let n be even Integer, m be odd Integer;
  assume A1: n <= m;
  consider k1 being Integer such that
    A2: n = 2*k1 by ABIAN:11;
  consider k2 being Integer such that
    A3: m = 2*k2+1 by ABIAN:1;
  per cases;
  suppose k1 <= k2;
    then 2*k1 <= 2*k2 by XREAL_1:64;
    hence n+1 <= m by A2, A3, XREAL_1:6;
  end;
  suppose k1 > k2;
    then k2 + 1 <= k1 by INT_1:7;
    then 2*(k2+1) <= 2*k1 by XREAL_1:64;
    then 2*k2+2 <= 2*k2+1 by A1, A2, A3, XXREAL_0:2;
    hence thesis by XREAL_1:6;
  end;
end;
