def prove(Eq):
n = Symbol.n(domain=Interval(2, oo, integer=True))
S = Symbol.S(dtype=dtype.integer * n)
x = Symbol.x(**S.element_symbol().dtype.dict)
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
given = [
ForAll[j:1:n - 1, x:S](Contains(
LAMBDA[i:n](Piecewise((x[0], Equality(i, j)),
(x[j], Equality(i, 0)), (x[i], True))), S)),
ForAll[x:S](Equality(abs(x.set_comprehension()), n))
]
Eq << apply(given)
Eq << discrete.combinatorics.permutation.adjacent.swap2.general.apply(
Eq[0])
Eq.permutation = discrete.combinatorics.permutation.adjacent.swapn.permutation.apply(
Eq[-1])
Eq << Eq.permutation.limits[0][1].this.definition
Eq << discrete.combinatorics.permutation.factorial.definition.apply(n)
Eq << Eq[-1].this.lhs.arg.limits_subs(Eq[-1].lhs.arg.variable,
Eq[-2].rhs.variable)
Eq <<= Eq[-1] & Eq[-2].abs()
F = Function.F(nargs=(), dtype=dtype.integer * n)
F.eval = lambda e: conditionset(x, Equality(x.set_comprehension(), e), S)
e = Symbol.e(dtype=dtype.integer)
Eq << Subset(F(e), S, plausible=True)
Eq << Eq[-1].this.lhs.definition
Eq << sets.subset.forall.imply.forall.apply(Eq[-1], Eq.permutation)
Eq.forall_x = ForAll(Contains(Eq[-1].lhs, F(e)),
*Eq[-1].limits,
plausible=True)
Eq << Eq.forall_x.definition.split()
P = Eq[-1].limits[0][1]
Eq << sets.imply.conditionset.apply(P)
Eq << Eq[-1].apply(sets.equality.imply.equality.permutation, x)
Eq.equality_e = Eq[-3] & Eq[-1]
Eq << sets.imply.conditionset.apply(F(e)).reversed
