This is a follow up to a question JLo asked. It has been recast
with subsets instead of colors.

Prove or disprove the following is true for all integers n ≥ 1:

    If Rⁿ is partitioned into n subsets S₁, S₂, ... , S_n; then
       ∃ i∈I_n ( |S_i| = R₊ )

Definitions and Nomenclature

I_n = { 1,2, ... , n }.

R is the set of real numbers ( complete ordered field ).

R₊ = { x∈R | x ≥ 0 }.

Rⁿ = { (x₁, x₂, ... , x_n) | x₁, x₂, ... , x_n∈R }.

Properties of S₁, S₂, ... , S_n:

   1) ∀ i∈I_n ( S_i ≠ Φ ),

   2) S₁ ∪ S₂ ∪ ... ∪ S_n = Rⁿ,

   3) ∀ i,j∈I_n ( i ≠ j ⇒ S_i ∩ S_j = Φ ),

   where Φ denotes the empty set.

If P,Q∈Rⁿ with P = (p₁,p₂, ... , p_n) and Q = (q₁,q₂, ... , q_n), then
   |PQ| = √[Sigma[(p_i - q_i)² ; i=1,n]]

If S ⊂ Rⁿ, then
   |S| = { |PQ|∈R₊ | P,Q∈S }.