U δ i , W δ j and V δ 2 be d -, ( d + 1)- and ( d + 2)-dimensional currents such that: To clarify the notation, we adopt the convention that variables with a δ subscript are chains on the simplicial complex K$_{δ}$ whereas a δ superscript merely indicates dependence on δ . Let K$_{δ}$ be any simplicial complex that triangulates P$_{δ}$ , X$_{δ}$ and S$_{δ}$ separately as well as the convex hull of their union. We may assume (applying the subdivision algorithm of Edelsbrunner and Grayson [18] and Theorem 3.3.6 if necessary) that the currents U$_{0}$ , U$_{1}$ , W$_{0}$ , W$_{1}$ , and W$_{2}$ can be pushed to K$_{δ}$ with expansion bound at most L and the maximum diameter Δ of a simplex of K$_{δ}$ satisfies Claim 3.3.7.1. F ( T ) ≤ lim$_{δ}$$_{↓}$$_{0}$ F$_{K}$$_{δ}$ ( P$_{δ}$ ) Proof of claim. By the triangle inequality and since any simplicial flat norm decompo- sition is a candidate decomposition for the flat norm, we have The claim follows from letting δ ↓ 0 and noting that F ( T − P$_{δ}$ ) → 0. Claim 3.3.7.2. F ( T ) = lim$_{δ}$$_{↓}$$_{0}$ F$_{K}$$_{δ}$ ( P$_{δ}$ ) Proof of claim. In light of Claim 3.3.7.1, we must show that F ( T ) ≥ lim$_{δ}$$_{↓}$$_{0}$ F$_{K}$$_{δ}$ ( P$_{δ}$ ). 74 F ( T ) ≤ F ( T − P$_{δ}$ ) + F ( P$_{δ}$ ) ≤ F ( T − P$_{δ}$ ) + F$_{K}$$_{δ}$ ( P$_{δ}$ ) . Δ ≤ δ max { 1 , M( ∂U δ $_{0}$) , M( ∂U δ $_{1}$) , M( ∂W δ $_{0}$) , M( ∂W δ $_{1}$) , M( ∂W δ $_{2}$) } . (3.19) P$_{δ}$ − T = U δ 0 + ∂W δ $_{0}$, F ( P$_{δ}$ − T ) = M( U δ $_{0}$) + M( W δ $_{0}$) , (3.18a) X − X$_{δ}$ = U δ 1 + ∂W δ $_{1}$, F ( X − X$_{δ}$ ) = M( U δ $_{1}$) + M( W δ $_{1}$) , (3.18b) S − S$_{δ}$ = W δ 2 + ∂V δ $_{2}$, F ( S − S$_{δ}$ ) = M( W δ $_{2}$) + M( V δ 2 ) . (3.18c)