Relativity of simultaneity example in Resnick My question is a follow-up to this question about simultaneity. I would have posted it as a comment to the replies for that question, but I wasn't allowed to.
When Resnick introduces relativity of simultaneity, he gives the following example (see figure): S & S' are two inertial frames with a relative velocity v, and each with its own synchronised clocks and meter sticks. Two events leave marks, at A & B in reference frame S and at A' & B' in reference frame S'. The observers in the two frames are located at O (equidistant from A,B) and O'(equidistant from A',B'), respectively. When the event happens at A, A' coincides with A, and when the event happens at B, B' coincides with B.

Resnick goes on to show that the events can't be simultaneous for both observers because, if the events are simultaneous for O, then O' will see the light pulses at slightly different times (viewed from the S frame).
I'm missing something in this argument: What basic inconsistency will arise if the events were simultaneous in both the frames? What will happen if the clocks in S at A,B show the same time for the events, and the clocks in S' at A', B' show the same time for the events? 
Also, a related question: Will the synchronised clocks of S' appear unsynchronised (to each other) to the observer in S? How will the observer in S check this?
 A: First, from the point of view of $O$.
The lightning strikes at points $A$ and $B$ happen simultaneously. Light propagates away from those points, and since $O$ is halfway between $A$ and $B$, the light fronts reach him at the same moment (equal distances and equal velocities gives equal times). 
Now, for things as $O'$ sees them. The important thing to remember is that by the postulates of Special relativity, $O'$ must measure the speed of light as being $c$. It doesn't matter that he's moving towards $B$ and away from $A$ - the light fronts must be moving at speed $c$. So the two light fronts have equal speeds, and they started at equal distances from him (remember that he says that the lightning bolts hit $A'$ and $B'$, which are equidistant from him). So they should take equal amounts of time to reach him. And they do - if you're just looking at the time intervals. But they don't reach him at the same moment of time. How can we resolve this? They must have started at different times. The lightning bolt at $B'$ must have struck earlier than the lightning bolt at $A'$, which is why the $B'$ light front reaches him first.
To clarify one point which may not be obvious: $O$ says the two light spheres are centered about $A$ and $B$, which are stationary with respect to him/her. $O'$ says that the light spheres are centered about $A'$ and $B'$, which are stationary with respect to him ($O'$). 
Regarding the synchronization question - whenever we talk about an inertial reference frame (say $S$), we actually mean a family of observers that all have the same motion. All the observers in the same inertial reference frame have their clocks synchronized. How do they do it? They know their distances from each other, and they know the speed of light is constant. They start with a specific observer at the (designated) origin, and he sends out light pulses every second. The observers who are $1\,\mbox{m}$ away from him know that it will take $\frac{1\,\mbox{m}}{3\times 10^{8}\mbox{m}/\mbox{s}}$ for the signal to reach them, and offset their clocks accordingly so that all their clocks are synchronized with the clock at the origin. And so forth.
When you ask about how the clocks look like from another reference frame (say $S'$), you have to specify - do you mean just one particular observer in $S'$ (say the observer at $O'$)? Or do you mean the family of observers in $S'$? The family of observers in $S'$ would state that the clocks in $S$ are all running slow at the same rate, so that they're synchronized. For a single observer, you'd have to take into account light travel times and the like - I'm not sure what the answer to that would be.
A: 
I'm missing something in this argument: what basic inconsistency will
  arise if the events were simultaneous in both the frames?

The one-way speed of light will not be c in all inertial frames of reference.
Essentially, the clocks at rest in each frame are synchronized according to the Einstein synchronization convention.
Indeed, the Lorentz transformation assumes Einstein synchronization.
Thus, Einstein synchronization guarantees that the one-way speed of light is c and thus, that simultaneity is relative - synchronized spatially separated clocks at rest in one frame are not synchronized according to a relatively moving reference frame.
This is most easily seen by direct application of the Lorentz transformation.
Let clock A be located at $x= 0$ and clock B be located at $x = 1$ in the unprimed reference frame and assume that the clocks are synchronized in that frame:
$$t_A = t_B$$
Assuming standard configuration, according to a relatively moving reference frame with velocity $v$, when clock A reads $t_A = 0$, clock B reads
$$t_B = \frac{v}{c^2}s $$
Thus, clocks A & B are not synchronized in relatively moving frame of reference.
A: What I find interesting is that one event can be seen as two events by another. Can it not ?
If one bolt of lightening struck the ground near the little red man, and the light expanded outward until eventually striking mirrors that are located at the ends of a train passing by, the light will eventually return to the little red man via these reflections. Thus he gets a second view of the lightning strike.
But to those aboard the train, the light did not reach to opposite ends at the same time, thus to them it seemed to be 2 separate lightening strikes.

Am I missing something ?
A: Before trying to address your questions let's discuss some general features of the figure and setup:

Two events leave marks, at A & B in reference frame S and at A' & B' in reference frame S'. The observers in the two frames are located at O (equidistant from A,B) and O'(equidistant from A',B').

Along with O and O' the named elements A, B, A', and B' should be considered ("locations of") observers, too, since each is supposed to determine (and remember) with whom they took part in coincidence events, and in which order. Foremost, they are all likewise identifiable participants of the setup.
(It is of course beyond the scope of this question how to determine which, if any, such participants were and remained at rest to each other, allowing to be attributed some particular "distance" from each other, and permitting the comparison of "distances".)

O' will see the light pulses at slightly different times (viewed from the S frame).

The figure, especially comparing pictures (2) and (4), shows that O' saw the two signal events of "A and A' meeting in passing" and of "B and B' meeting in passing" not in coincidence (i.e. unlike O saw these two signal events) but first "B and B' meeting in passing" and afterwards "A and A' meeting in passing". (Indeed, inbetween O' observed the coincidence (in passing) with B; if the figure is meant to be so precise, and if $\frac{v}{c} = \beta \gt \frac{1}{2}$.)
These determinations (or setup presciptions) are not subject to any secondary arbitrary "view" such as "from the S frame", but they are proper and unambiguous.
However, the individual pictures in Resnicks's figure are apparently suggestive of the "perspective from the S frame", or rather: with particular attention to the relations between participants A, O, and B (instead of the relations between participants A', O', and B'). For instance:     
The first picture shows the three coincidence events of 


*

*"A and A' meeting in passing",

*"O and O' meeting in passing", and

*"B and B' meeting in passing"


together. And, consistent with the overall setup presciption (if I understand it correctly), there is something in common (which suggests putting these three coincidence events in the same picture):


*

*A's indication of "being passed by A'" and

*O's indication of "being passed by O'" and

*B's indication of "being passed by B'" 


were (mutually) simultaneous each other.
But in turn, 


*

*A''s indication of "being passed by A" and

*O''s indication of "being passed by O" and

*B''s indication of "being passed by B" 


were (pairwise) not simultaneous each other.
The other pictures are therefore presumably also meant to illustrate corresponding relations between participants A, O, and B. Since, according to the overall setup presciption, 


*

*O's indication of "being passed by A'" and

*B's indication of "being passed by O'" 


were simultaneous to each other,
therefore picture 4 (with $\beta \gt \frac{1}{2}$) displays consistently some particular indication of O after having been passed by A'.
In contrast, if $\frac{\sqrt{5} - 1}{2} \gt \beta \gt \frac{1}{2}$ then 


*

*O''s indication of "seeing the coincidence event of A and A' having met in passing" 


was before


*

*A''s indication of "being passed by O";


therefore picture (4) is certainly not representative of relations between participants A', O', and B', for all values $\beta \gt \frac{1}{2}$.

if the events are simultaneous for O

Simultaneity is not defined for entire events (which in general have many different participants who coincide at either one of these events);
but simultaneity (or otherwise: "temporal order") is defined for indications of certain pairs participants at events (namely those who were and remained at rest to each other).
To repeat the conclusions concerning simultaneity (or "temporal order") for the two signal events of the setup given above:


*

*A's indication of "being passed by A'" and

*B's indication of "being passed by B'" 


were simultaneous each other. And:


*

*A''s indication of "being passed by A" and

*B''s indication of "being passed by B" 


were (pairwise) not simultaneous each other; indeed


*

*B''s indication of "being passed by B" had been before

*A''s indication of "being passed by A".


But simultaneity (or dis-simultaneity) is not attributable to the entire two signal events


*

*"A and A' meeting in passing", and

*"B and B' meeting in passing".


Now to your concrete questions (in an order which I find convenient):

What will happen if the clocks in S at A,B show the same time for the events, and the clocks in S' at A', B' show the same time for the events? 

Then such clocks associated with A' and B' would be called not synchronized.

Will the synchronised clocks of S' appear unsynchronised (to each other) to the observer in S? 

Having been and remained synchronized (or un-synchronized) is a proper attribute of any given system of clocks (which were and remained at rest to each other) themselves; it's not a matter of "appearance" to anyone who does not belong to that system.

How will the observer in S check this?

At least in principle anyone would consult the observations and (proper) conclusions obtained by the members of the system under consideration themselves. The required observational data, namely indentifying who took part together in coincidence events, and who saw which signal events in which order, is presumably so basic as to be unambiguously comprehensible by anyone who is conscientious enough to wonder "how to check?".

what basic inconsistency will arise if the events were simultaneous in both the frames? 

First of all note again that simultaneity is a matter of indications of individual participants in events, not a matter of entire events.
If we require (different from the conclusions listed above) that 


*

*A''s indication of "being passed by A" and

*B''s indication of "being passed by B" 


were simultaneous to each other, and also (as above) that


*

*A's indication of "being passed by A'" and

*B's indication of "being passed by B'" 


were simultaneous each other, too, then, yes, this requirement is inconsistent with the setup presciption given above (if I understand it correctly). 
(If one setup condition could be dropped or modified, in order to restore consistency with the "new requirements", then the "direction of motion, $\vec v$" might be changed from "along A, O, and B" and likewise in turn "along B', O', and A'" into "perpendicular motion". Accordingly, the first picture could remain; but the other three would no longer apply at all.) 
