# Question:In Maple, how do I show two geometric regions (defined by some formulae) are equal?

## Question:In Maple, how do I show two geometric regions (defined by some formulae) are equal?

Maple 2022

Let us begin with few simulations:

 > restart;
 >
 memory used=0.57TiB, alloc change=91.51MiB, cpu time=18.77m, real time=15.86m, gc time=4.91m
 >
 memory used=0.56TiB, alloc change=-12.08MiB, cpu time=18.70m, real time=15.11m, gc time=5.69m

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 sympy.vector.)
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?

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