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

theorem
  y <> +infty & 0 < x & 0 < y implies 0 < x / y
  proof
    assume that
A1: y <> +infty and
A2: 0 < x and
A3: 0 < y;
    per cases by XXREAL_0:14;
    suppose x in REAL;
      then reconsider x1 = x as Element of REAL;
      reconsider y1 = y as Element of REAL by A1,A3,XXREAL_0:14;
      x/y = x1/y1 by EXTREAL1:2;
      hence thesis by A2,A3;
    end;
    suppose x = +infty;
      hence thesis by A1,A3,XXREAL_3:83;
    end;
    suppose x = -infty;
      hence thesis by A2;
    end;
  end;
