FUNCTION :  findtype9 - find type 9 identities
                

CALLING SEQUENCE :  findtype9()
                    

PARAMETERS :  NONE
                          

GLOBAL VARIABLES : 
        xprint,NEWJACID,RJID,SYMJID,
        TT1,TT2,PROVEDFL9,EBL

SYNOPSIS :   
        Before running the functions G,H,GM,HM,GE,HE must be defined.   

        findtype9(T) cycles through symbolic expressions
        
           _G(1)^a_H(1)^b -  _H(1)^a_G(1)^b + x
        
        where  x is 0 or -1, with a, b smallest such positive integers,
        using CHECKRAMIDF to check whether the expression corresponds to a 
        likely eta-product.
        If proveit is true then provemodfuncidBATCH (from theataids 
        package) is used to prove it.  The procedure also returns a list 
        of [a,b,x] which give identities.

EXAMPLES :   

>  with(qseries):
>  with(thetaids):
>  with(ramarobinsids):
>  xprint:=false: proveit:=true:
>  G:=j->1/GetaL([1],5,j):H:=j->1/GetaL([2],5,j):
>  GM:=j->1/MGetaL([1],5,j): HM:=j->1/MGetaL([2],5,j):
>  GE:=j->-GetaLEXP([1],5,j):HE:=j->-GetaLEXP([2],5,j):
>  findtype9(12);
*** There were NO errors.  Each term was modular function on
    Gamma1(5). Also -mintotord=2. To prove the identity
    we need to  check up to O(q^(4)).
    To be on the safe side we check up to O(q^(12)).
*** The identity below is PROVED!
[11, 1, 1]
                                                        6
            11              11             11 eta(5 tau) 
       _G(1)   _H(1) - _H(1)   _G(1) - 1 = --------------
                                                     6   
                                             eta(tau)    
                          [[11, 1, 1]]


DISCUSSION :
    For G,H defined 
    G(1)^11 H(1) - H(1)^11 G(1) - 1, is an eta-product and the 
    identity is proved.        

SEE ALSO :  

findtype1, findtype2,
findtype3, findtype6,
findtype6, findtype6,
findtype7, findtype8,
findtype9, findtype10