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

**R**is partitioned into n subsets S

^{n}_{1}, S

_{2}, ... , 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 }.

**R**= { (x

^{n}_{1}, x

_{2}, ... , x

_{n}) | x

_{1}, x

_{2}, ... , x

_{n}∈

**R**}.

Properties of S

_{1}, S

_{2}, ... , S

_{n}:

1) ∀ i∈I

_{n}( S

_{i}≠ Φ ),

2) S

_{1}∪ S

_{2}∪ ... ∪ S

_{n}=

**R**,

^{n}3) ∀ i,j∈I

_{n}( i ≠ j ⇒ S

_{i}∩ S

_{j}= Φ ),

where Φ denotes the empty set.

If P,Q∈

**R**with P = (p

^{n}_{1},p

_{2}, ... , p

_{n}) and Q = (q

_{1},q

_{2}, ... , q

_{n}), then

|PQ| = √[Sigma[(p

_{i}- q

_{i})

^{2}; i=1,n]]

If S ⊂

**R**, then

^{n}|S| = { |PQ|∈

**R**| P,Q∈S }.

_{+}