reserve x,y,z,w for ExtReal,
  r for Real;
reserve f,g for ExtReal;
reserve t for ExtReal;

theorem
  x < y & z < 0 & z <> -infty implies y/z < x/z
proof
  assume that
A1: x < y and
A2: 0 > z and
A3: z <> -infty;
  per cases by A3,XXREAL_0:14;
  suppose
    z = +infty;
    hence thesis by A2;
  end;
  suppose
    z in REAL;
    then reconsider z as Real;
    z" < 0 by A2;
    hence thesis by A1,Th102;
  end;
end;
