Without it, if T2 urns on and directs the current to T4, it may not pull 
down the base , and in effect the voltage between its emitter and base, 
of T3 low enough to turn it off.
UBE(T2)+UBE(T4) must be < UCE(T4)+UBE(T4) and may not be the case.
But UBE(T2)+UBE(T4) < UCE(T4)+Uf(D2)+UBE(T4) cetainly holds.

I'm not sure that I understand what you said . I'm kinda lost with the 
UBE UCE and UF symbols because I'm not familiar with them.
Have a look at the screenshots that I made with/without Diode and tell 
me what I'm doing wrong .
Plus I'd like to know what's UBE UCE and UF with the use of the voltage 
shown in my screenshots . That will be great to understand their 
meanings.

As you can see in the screenshots . They both lead me to the same 
result. That's why I got confused

Alex wrote:
> They both lead me to the same
> result.

Why do you write 0V an the first output an 0.2V on the second ?

Why is Uf just 0.2V ?

Use realistic UBE, UCE and Uf values, for instance 0.7 0.2 and 0.7V or 
from a transistor data sheet or use LTSpice.

The input transistor also has wrong 0.7V on base. It mainly works like 2 
diodes, either directing the current thru the 4k7 resistor into the base 
of the T2 or into te input.

I made a mistake in the formuals

UBE(T2)+UBE(T4) must be < UCE(T4)+UBE(T3) and this will not be the case.
But UBE(T2)+UBE(T4) < UCE(T4)+Uf(D2)+UBE(T3) cetainly holds

> They both lead me to the same result.

Except that in the right diagramm T3 and T4 are both on and you have a 
short.  In the left, the 0.2V is not enough for the diode to conduct[1] 
and no current flows through T3.


[1] especially as the output is also 0.2V and not 0 (the 
collector-emitter voltage of T4) and hence the voltage across the diode 
is ~0V.

>Except that in the right diagramm T3 and T4 are both on and you have a
>short.  In the left, the 0.2V is not enough for the diode to conduct[1]
>and no current flows through T3.

So the diode D2 function would be on this case : Preventing a short 
circuit ?

Alex wrote:
> So the diode D2 function would be on this case : Preventing a short
> circuit ?

Yes, or in other words: allow the output circuit to work as intended.