Question: Is there a underlying routine of showing that one geometric region which has been decribed implicitly (but not parametrically) is included in another?

Let a, b, c, d be real numbers. Define two geometric regions implicitly: a＋b＋c＋d＞0∧ab＋ac＋ad＋bc＋bd＋cd＞a²＋b²＋c²＋d², and abcd＞0∧abc＋abd＋acd＋bcd＞0. Suppose that somebody wants to prove that the former region is a subset of the latter one. Can we implement such an implication simply using certain ready-made commands in basic Maple? In other words, it is hoped that 

restart;
RealDomain:-simplify(forall([a, b, c, d], a + b​​​​​​​ + c + d > 0 and a*b + a*c + a*d + b*c + b*d + c*d > a^2 + b^2 + c^2 + d^2 implies a*b*c*d > 0 and a*b*c + a*b*d + a*c*d + b*c*d > 0));

Unfortunately, simplify just returns the statement unevaluated. So, how to address it in internal Maple? 

Note. The desired result (or return value) is true (see below).

1. restart; # Do not with(Logic): here
   RegularChains:-SemiAlgebraicSetTools:-QuantifierElimination(&A [d, c, b, a], ((d + c + b + a > 0) &and (b*a + c*a + d*a + c*b + d*b + d*c > a^2 + b^2 + c^2 + d^2)) &implies ((d*c*b*a > 0) &and (d*c*b + d*c*a + d*b*a + c*b*a > 0)));

2. restart;
   not SMTLIB:-Satisfiable(`not`(a + b + c + d > 0 and a*b + a*c + a*d + b*c + b*d + c*d > a^2 + b^2 + c^2 + d^2 implies d*c*b*a > 0 and a*b*c + a*b*d + a*c*d + b*c*d > 0)); # assuming real 

​​​​​​​​​​​​​​​​​​​​​Either ⒈ or ⒉ needs external calls. It seems that the Maple standard library functions in List of Commands still fail in simplifying expressions in the aforementioned form. Am I right?