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U δ i , W δ j and V δ 2 be d -, ( d + 1)- and ( d + 2)-dimensional currents such that: 

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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 

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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$_{δ}$ ) 

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Proof of claim. In light of Claim 3.3.7.1, we must show that F ( T ) ≥ lim$_{δ}$$_{↓}$$_{0}$ F$_{K}$$_{δ}$ ( P$_{δ}$ ). 

