read `EKHAD.txt`:

t0:=time():

F1:=(-2)^n0*binomial(2*n+2*m,n0)*binomial(6*n-j-n0,3*n):
F2:=(-1)^(n+m)*(-1)^(k)*binomial(2*n+2*m,n+m+k)*binomial(4*n-2*m-j,2*n-m-k):

print(`Proof of the identity at the bottom of page 7 of the Zeilberger-Zudilin paper`):
print(``):
print(`by Shalosh B. Ekhad `):
print(``):

print(`Proposition`):
print(``):
print(`Let F1 be `):
print(``):
print(F1):
print(``):
print(`Let F2 be `):
print(``):
print(F2):
print(``):
print(`The sum of F1 with resepect to n0 equals the sum of F2 with respect to k`):
print(``):
print(`We will prove that both sums satisfy the same second-order linear recurrences with respect to each of the three`):
print(``):
print(`discrete variables m,n,j`):
print(``):
print(`-----------------------------`):
print(``):
print(`For the variable m, and F1,  we have the following lemma`):
print(``):
zeilpap(F1,n0,m):
print(``):

print(`For the variable m, and F2,  we have the following lemma`):
print(``):
zeilpap(F2,k,m):
print(``):

print(`One can see that these are recurrences are identical`):
print(``):


print(`--------------------------------------------`):
print(``):
print(`For the variable n, and F1,  we have the following lemma`):
print(``):
zeilpap(F1,n0,n):
print(``):

print(`For the variable n, and F2,  we have the following lemma`):
print(``):
zeilpap(F2,k,n):
print(``):

print(`One can see that these are recurrences are identical`):
print(``):
print(`--------------------------------------------`):
print(``):

print(`--------------------------------------------`):
print(``):
print(`For the variable j, and F1,  we have the following lemma`):
print(``):
zeilpap(F1,n0,j):
print(``):

print(`For the variable j, and F2,  we have the following lemma`):
print(``):
zeilpap(F2,k,j):
print(``):

print(`One can see that these are recurrences are identical`):
print(``):
print(`--------------------------------------------`):
print(``):



print(`This finishes this article that took`, time()-t0, `seconds.`):

quit: