Proof. We apply properties of norms to obtain F (( αT$_{n}$ + βU$_{n}$ ) − ( αT + βU )) ≤ F ( αT$_{n}$ − αT ) + F ( βU$_{n}$ − βU ) = | α | F ( T$_{n}$ − T ) + | β | F ( U$_{n}$ − U ) . Letting n → ∞ yields the linearity result. Now let X$_{n}$ and S$_{n}$ be m - and ( m +1)-currents such that X$_{n}$ + ∂S$_{n}$ is a flat norm decomposition of T$_{n}$ − T for n = 1 , 2 , . . . , observing that The boundary result follows in the limit. In the case of the simplicial flat norm, an input integral chain is guaranteed an integral chain decomposition whenever the simplicial complex is totally unimodular [24]. This occurs when the complex is free of relative torsion which is the case for any ( d + 1)-complex in R d +1 or when triangulating a compact, orientable ( d + 1)-dimensional manifold. Theorem 3.2.3 (Simplicial flat norm integral decomposition [24]) . If K is a simplicial ( d + 1) -complex embedded in R d $^{+1}$, then for any integral d -chain P on K , the optimal simplicial flat norm value for P is attained by an integral decomposition. We state the simplicial deformation theorem and sketch a portion of its proof. We will later modify it to obtain a multiple current deformation theorem that preserves linearity (Theorem 3.3.1). Theorem 3.2.4 (Simplicial deformation theorem [24]) . Suppose K is a p -dimensional simplicial complex in$_{R}$ q and T is a normal d -current supported on the underlying space of K . There exists a simplicial d -current P supported on the d -skeleton of K with boundary 59 F ( ∂T$_{n}$ − ∂T ) = F ( ∂ ( X$_{n}$ + ∂S$_{n}$ )) = F ( ∂X$_{n}$ ) ≤ M( X$_{n}$ ) ≤ F ( T$_{n}$ − T ) .