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

theorem
  x < y & 0 < z & z <> +infty implies x/z < y/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,Th72;
  end;
end;
