FUNCTION :  findtype3 - find type 3 identities
                

CALLING SEQUENCE :  findtype3(T)
                    

PARAMETERS :   T - positive integer  
                  -            

GLOBAL VARIABLES : 
        xprint,NEWJACID,RJID,SYMJID,
        TT1,TT2,PROVEDFL1,EBL

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

       findtype3(T) cycles through symbolic expressions
       
                  _G(a[1]) _G(b[1]) + c[1] _H(a[1]) _H(b[1])
                  ------------------------------------------
                  _G(a[2]) _H(b[2]) + c[2] _H(a[2]) _G(b[2])
       where  2 ≤ n ≤ T, a_1b_1=a_2b_2=n, (a_1,b_1,c_1,d_1)=1,  a_1 \le b_1, 
       b_2 < a_2, c_1,c_2 in {-1,1},  and
       
       (*)
          GE(a[1]) + GE(b[1]) - (HE(a[2]) + HE(b[2])) in Z,
          GE(a[2]) + GE(b[2]) - (HE(a[2]) + GE(b[2])) in Z,
       
       and [a_2,b_2,c_2] is not an element for the list {myramtype1
       (product earlier by findtype1), 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_1,b_1,c_1,a_2,b_2,c_2]
       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):
>  findtype3(14);
"n=", 10
*** There were NO errors.  Each term was modular function on
    Gamma1(70). Also -mintotord=96. To prove the identity
    we need to  check up to O(q^(98)).
    To be on the safe side we check up to O(q^(236)).
*** The identity below is PROVED!
[1, 14, 1, 7, 2, -1]
                                             2           2
     _G(1) _G(14) + _H(1) _H(14)   eta(7 tau)  eta(2 tau) 
     --------------------------- = -----------------------
      _G(7) _H(2) - _H(7) _G(2)               2         2 
                                   eta(14 tau)  eta(tau)  
                     [[1, 14, 1, 7, 2, -1]]


DISCUSSION :
    For G,H defined 

     G(1) G(14) + H(1) H(14)   
     -----------------------                               
     G(7) H(2)  - H(7) G(2)              

    is an eta-product and the identity is proved.

SEE ALSO :  

findtype1, findtype2,
findtype3, findtype4,
findtype5, findtype6,
findtype7, findtype8,
findtype9, findtype10