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

theorem Th5:
  y <> +infty & x < 0 & 0 < y implies x / y < 0
  proof
    assume that
A1: y <> +infty and
A2: x < 0 and
A3: 0 < y;
    reconsider y1 = y as Element of REAL by A3,A1,XXREAL_0:14;
    per cases by XXREAL_0:14;
    suppose x in REAL;
      then reconsider x1 = x as Real;
      x/y = x1/y1 by EXTREAL1:2;
      hence thesis by A2,A3;
    end;
    suppose x = +infty;
      hence thesis by A2;
    end;
    suppose x = -infty;
      hence thesis by A1,A3,XXREAL_3:86;
    end;
  end;
