@Carl Love Ok so I see now.  Just tested it this way

(f, (E, V) -> V)(expand(A), exp(T))  gives  f(A, exp(T)), exp(T)

@Christian Wolinski Very nice solution. I don't quite get the technicality of the (E,V)->V So E=expand(A) and V =exp(T) in this case and somehow (solve,((E,V)->V))(expand(A),exp(T))  includes exp(T) in the answer.   I notice the " , " after solve and the enclosing (....) are necessary.  I did check through help but didn't see an explination. Of course maybe I missed that.

Edit:- I see that (E,V)->V))(expand(A),exp(T)   is eqivalent to 

f := (E, V) -> V;

f(expand(A), exp(T));

@Carl Love Can you explain what index means in relation to solve? 

@Carl Love You are correct.  It took me a while to remenber the syntax to use.

 Interesting problem.  So I gather there are "s" nested summation loops. I asked a question a few months ago that required with a variable number of summation loops. @Carl Love answered it.

Is it possible to Add/sum by mapping multiple values from a list - MaplePrimes

So just to clarify can t[j] and t[i]  be replaced with T[j] , T[i] to totaly distinguish them from t?

I presume the inputs are M and s.

 Edit:-On problems like this a I start with small values for s and M and build the nested summations/products to handle them before trying for the total solution. I attach  document to see if my begining intrepertations are correct.

Download 28-3-22_A_repetative_summation.mw

@Ali2020 The document stated "I do not know how you achieved your resuls". As they are listed vertically they could come from a loop. That is all.

@max125 Opps.. I post my revised answer here. Makes the answer simpler.

Download 23-3-22_A1_integration.mw

@nm I would just install over it.  Only when I have a big problem do I "clean" a program out of a computer.

@dharr Oh.  I didn't realist that about view. Very simple.

@vv Yes I hit that. Couldnt fix it so I copied and pasted the answer.

@Carl Love Answer is much neater.

@dharr , @tomleslie   Thanks. I had tried Null. Forgot about the all caps.

@Carl Love That is good. My old attempt cripples the graphics on the computer.

@acer Excellent diagnosis That looks like what happened.  I will fix it during the week and let you know.

So, in this case ooes the "DirectSearch.mla" now contain the DirectSearch, RationalTrigonometry and RonanRoutines packages?

@Joe Riel  Thank you. I don't understand whay this result is returned. The screen shot shows what is in the returned location.

See attached file on what I did.  AFIK. the mla files are in different locations; based on what I could find. I gave the different folders' locations in the question. 
 

Download Find_Library.mw

@mmcdara Thank you. Very detailed.  The matrix is built of 3 projective points [x,y,z]. Well I was trying to see if the determinalt =0. That would imply the points are colinear. I was trying last night to normailse them to [x/z,y/z,1], but even that was to much. If I could that would reduce the determinant multiplication greatly. e.g 

det=a_11*(a22-a23)-a12*(a21-a23)+a13*(a21-a22)

I will check tonight and see if the z points have a common factor.

I totally understand your point about unreadable hugh expressions.