from proveit import Etcetera
from proveit.logic import Forall, InSet, Equals, NotEquals, Implies
from proveit.number import Integers, NaturalsPos, Reals, RealsPos, Complexes
from proveit.number import Divide, frac, Add, Sub, Sum, Mult, Exp
from proveit.common import a, b, c, n, w, x, y, z, P, S, xMulti, wEtc, xEtc, yEtc, zEtc, PyEtc
from proveit.number.common import zero, one, ComplexesSansZero
from proveit import beginTheorems, endTheorems
beginTheorems(locals())
divideRealClosure = Forall([a, b],
InSet(Divide(a, b), Reals),
domain=Reals,
conditions=[NotEquals(b, zero)])
divideRealClosure
divideRealPosClosure = Forall([a, b],
InSet(Divide(a, b), RealsPos),
domain=RealsPos,
conditions=[NotEquals(b, zero)])
divideRealPosClosure
fractionRealClosure = Forall([a, b],
InSet(frac(a, b), Reals),
domain=Reals,
conditions=[NotEquals(b, zero)])
fractionRealClosure
fractionPosClosure = Forall([a, b],
InSet(frac(a, b), RealsPos),
domain=RealsPos,