def prove(Eq):
a = Symbol.a(real=True)
b = Symbol.b(real=True, domain=Interval(a, oo, left_open=True))
assert b > a
assert b - a > 0
f = Function('f')
given = Equality.continuity(f, a, b)
z = given.lhs.args[1]
Eq << apply(given)
m = Symbol.m(definition=MIN(f(z), (z, a, b)))
M = Symbol.M(definition=MAX(f(z), (z, a, b)))
Eq.min, Eq.max = m.equality_defined(), M.equality_defined()
Eq << axiom.calculus.integral.intermediate_value_theorem.apply(given)
Eq.intermediate_value = Eq[-1].this.limits[0][2].subs(
Eq.max.reversed).this.limits[0][1].subs(Eq.min.reversed)
Eq << m.definition.assertion().integral(
(z, a, b)).subs(Eq.min.reversed) / (b - a)
Eq << M.definition.assertion().integral(
(z, a, b)).subs(Eq.max.reversed) / (b - a)
Eq << (Eq[-1] & Eq[-2])
# Eq << (Eq[-1].reversed & Eq[-2])
# Eq << (Eq[-1] & Eq[-2].reversed)
# Eq << (Eq[-1].reversed & Eq[-2].reversed)
# Eq << (Eq[-2] & Eq[-1])
# Eq << (Eq[-2].reversed & Eq[-1])
# Eq << (Eq[-2] & Eq[-1].reversed)
# Eq << (Eq[-2].reversed & Eq[-1].reversed)
Eq << Eq.intermediate_value.subs(Eq.intermediate_value.rhs, Eq[-1].lhs)
Eq << (Eq[-1] & Eq[-2])
Eq << Eq[-1].split()
Eq << Eq[-1] * (b - a)
Eq << Eq[-1].this.function.rhs.ratsimp().reversed
def prove(Eq):
n = Symbol.n(integer=True)
d = Symbol.d(integer=True)
Eq << apply(n, d)
M = Symbol.M(shape=(n, n), definition=Eq[0].rhs.args[0].arg)
Eq << M.this.definition
Eq << Eq[0].subs(Eq[-1].reversed)
Eq << Eq[-1][0]
Eq << Eq[2][0]
Eq << Eq[-2].subs(Eq[-1])
