Let us begin with few simulations: 
 

Download iDistributionVector.mws

Well, I'd like to prove (through the use of Maple): 

transform((x1, x2, x3, x4, x5) -> [x1 - x3, x2 - x4, x5])(inequal(And((x || (1 .. 5)) >=~ 0, norm([x || (1 .. 5)], 1) = 1))) # not Maple syntax

is equivalent to a filled pyramid

ImplicitRegion((X, Y, Z), 0 <= Z <= 1 - abs(X) - abs(Y)) # not SymPy syntax

, 

transform((x1, x2, x3, x4, x5) -> [x1 - x3, x2 - x4, x5])(inequal(And((x || (1 .. 5)) >=~ 0, norm([x || (1 .. 5)], 2) = 1))) # not Maple syntax

is equivalent to a hemi-ball

ImplicitRegion((X, Y, Z), 0 <= Z <= sqrt(1 - X**2 - Y**2)) # not SymPy syntax

, and 

transform((x1, x2, x3, x4, x5) -> [x1 - x3, x2 - x4, x5])(inequal(And((x || (1 .. 5)) >=~ 0, norm([x || (1 .. 5)], 'infinity') = 1))) # not Maple syntax

is equivalent to a solid cuboid

ImplicitRegion((X, Y, Z), -1 <= X <= 1 & -1 <= Y <= 1 & 0 <= Z <= 1) # not SymPy syntax

. (Here, for the convenience of the descriptions, I utilize some non-standard notation from .) 
Note that ”two regions are equal" is a two-way property, which means the following proof 

is(Z >= 0) and is(Z <= 1 - abs(X) - abs(Y)) assuming (X, Y, Z) =~ (x1 - x3, x2 - x4, x5), x || (1 .. 5) >=~ 0, add(x || (1 .. 5)) = 1;
                              true
(*Accordingly, the latter region is a subset of the former one.*) 

is incomplete (because it's hard to determine whether the is routine always performs equivalent transformations in internal evaluation). 

So, can I execute such eliminations in Maple?

After studying the plottools:-transform command, I intend to visualize the following regions with constrained parameters in : 
 

But Mma gives 

The first instance (with default settings) is the same, but as for the second instance, which graph is correct? 

restart;
with(plottools):
with(plots):
transform((u, v) -> [u^3 - v^2, u^2 - v^3])(inequal(Or(u^2 + 4*v^2 <= 4, And(u^2 + v^2 < 4, (u + 2)^2 + 2*(v - 1)^2 <= 2)), nolines));
transform((s, t) -> [s^2*sqrt(t)*cos(t), s^2*sin(t)])(inequal(`and`(1 <= s, 5*s <= 5 + t, t < 5), s = 1 .. 2, t = 0 .. 5));

Download TransformedRegion.mws

The range is wrong. For details, see below, please.
 

Download SmartPlots.mw

The help page claims that smartplot(..) will call 2-D plot procedures ultimately, but why is the smartplots command incompatible with the use of assume?

Mathematica's Dimensions returns a list of the allowed levels, which has been implemented in Maple as . But 

MmaTranslator:-Mma:-Dimensions(<<1 | 2>, <1 | 0>>);

returns [6], and 

MmaTranslator:-Mma:-Dimensions([[1, 2], [1, 0]]);

returns [2, 2, 4]. What happened here?

It should be [2, 2].

Download Mma[Dimensions].mws

Execute the following codes in Maple input (1-D math) in the Standard interface. 

(cat("A".."C"),cat("d".."f"))||'$"G".."I",$"j".."l"'; 

Then an error occurred. But if one copy them into 2-D math (instead of Convert To>2-D Math Input) and execute them directly, everything goes without any error messages.

It says that mixed 1-D and 2-D math inside one command is not supported and not recommended stylistically. However, I just want to understand the reason why an error is raised here. 

Download unexpectedConcatenation.mws