reserve x, y, z, s for ExtReal;
reserve i, j for Integer;
reserve n, m for Nat;
reserve x, y, v, u for ExtInt;

theorem Th10:
  x < y implies x + 1 <= y
  proof
    assume
A1: x < y;
    per cases;
    suppose
      x in REAL & y in REAL;
      then reconsider x1 = x, y1 = y as Real;
      ex p, q being Real st p = x+1. & q = y & p <= q
      proof
        take x1+1, y1;
        thus x1+1 = x+1. by XXREAL_3:def 2;
        thus y1 = y;
        thus x1+1 <= y1 by A1,INT_1:7;
      end;
      hence thesis;
    end;
    suppose not x in REAL or not y in REAL;
      then x = +infty or x = -infty or y = +infty or y = -infty by XXREAL_0:14;
      hence thesis by A1,XXREAL_0:3;
    end;
  end;
