Could s.o. help me with Verifying this system of differential equation

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I would like to verify that these functions :

F1=(sqrt(3)*sinh(sqrt(3)*x)+3*sin(x))/(3*cos(x)+4-cosh(sqrt(3)*x))
F2=(2*sqrt(3)*sin((1/2)*x)*sinh((1/2)*sqrt(3)*x)+6*cos((1/2)*x)*cosh((1/2)*sqrt(3)*x))/(3*cos(x)+4-cosh(sqrt(3)*x))
G1=(4*sqrt(3)*cos((1/2)*x)*sinh((1/2)*sqrt(3)*x))/(3*cos(x)+4-cosh(sqrt(3)*x))
G2=(6*cos((1/2)*x)*cosh((1/2)*sqrt(3)*x)+2*sqrt(3)*sin((1/2)*x)*sinh((1/2)*sqrt(3)*x)+2-2*cos(sqrt(3)*x))/(3*cos(x)+4-cosh(sqrt(3)*x))

are a solution to the system of differential equations:

diff(F1) == F2^2
diff(F2) == F1*F2+G1
diff(G1) == F2*(F2-G2)+F2
diff(G2) == F2*(F1-G1)+G1

Now, i have tried to solve the system directly with "desolve_system" which obviously didn't work. After that i just differentiated the functions above using sage and hoped that sage could somehow verify that e.g F1'=F2^2 but sadly i couldn't show that this was equal because they had such different forms that sage didn't recognize them as equal. But i know for a fact that these functions satisfy the system of DE. is there any way i could prove this using sage, and if so: how could i do it? and i am looking for an algebraic solution not a numeric one.

I would appreciate any answert to my post