def prove(Eq):
i = Symbol.i(integer=True, nonnegative=True)
X = Symbol.X(shape=(oo, ), distribution=NormalDistribution(0, 1))
assert X[i].is_extended_real
assert X.is_random
k = Symbol.k(integer=True, positive=True)
Y = Symbol.Y(distribution=ChiSquaredDistribution(k))
assert Y.is_extended_real
assert Y.is_random
Eq << apply(X, Y)
Y = Symbol.Y(shape=(oo, ), definition=LAMBDA[k](Sum[i:k](X[i] * X[i])))
assert Y.is_nonnegative
assert Y.is_finite
Eq << Y.equality_defined() # 1
Eq << Eq[0].subs(Eq[-1].reversed) # 2
Eq << Eq[-1].subs(k, k + 1) # 3
Eq << Eq[1].subs(k, k + 1) - Eq[1] + Y[k] # 4
Eq.x_squared_y = Eq[-2].subs(Eq[-1]) # 5
Eq << Eq.x_squared_y.lhs.this.doit(evaluate=False)
Eq << Eq[-1].this.rhs.args[3].function.args[-1].doit(deep=False)
(_y, *_), *_ = Eq[-1].rhs.args[-1].limits
y = Eq[0].lhs.symbol
assert y.is_nonnegative
Eq << Eq[0].subs(y, _y)
Eq << Eq[-2].subs(Eq[-1])
Eq << Eq[-1].subs(Eq.x_squared_y)
Eq << Eq[-1].this.lhs.expand()
from sympy import sin
t = Symbol.t(domain=Interval(0, pi / 2))
assert t.is_zero is None
Eq << Eq[-1].this.rhs.args[-1].limits_subs(_y, y * sin(t)**2)
Eq << Eq[-1].this.rhs.args[-1].function.powsimp()
Eq << Eq[-1].solve(Eq[-1].rhs.args[-1])
Eq << wallis.apply(1, k)
(x, *_), *_ = Eq[-1].lhs.limits
(t, *_), *_ = Eq[-2].lhs.limits
Eq << Eq[-1].this.lhs.limits_subs(x, t)
# expand the BETA function into gamma function
Eq << Eq[-1].this.rhs.expand(func=True)
Eq << Eq[2].subs(k, 1)
Eq << Eq[1].subs(k, 1).doit(deep=False)
Eq << Eq[-2].subs(Eq[-1])
Eq << Eq[-1].lhs.this.doit(evaluate=False)
