Recall that X + ∂S = T is an optimal flat norm decomposition of T and X$_{δ}$ and S$_{δ}$ are polyhedral approximations to X and S on our simplicial complex K$_{δ}$ . Using the decompositions in Equations (3.16) and (3.18), we can write: Now apply Theorem 3.3.6 with ϵ = 1 to the currents U δ i and W δ j for all i ∈ { 0 , 1 } and j ∈ { 0 , 1 , 2 } to obtain U$_{i,δ}$ and W$_{j,δ}$ on the simplicial complex K$_{δ}$ with Applying Equations (3.17) to (3.19), we obtain the following from Equation (3.21): We apply the linearity result of Theorem 3.3.6 to Equation (3.20) along with the fact that P$_{δ}$ , X$_{δ}$ , and ∂S$_{δ}$ are fixed by projection to the d -skeleton of K$_{δ}$ to yield which, as all quantities are chains on K$_{δ}$ , is a candidate simplicial flat norm decom- position of P$_{δ}$ . Using this observation, the triangle inequality, and Equations (3.17) 75 P$_{δ}$ = ( X$_{δ}$ + U$_{0}$$_{,δ}$ + U$_{1}$$_{,δ}$ ) + ∂ ( S$_{δ}$ + W$_{0}$$_{,δ}$ + W$_{1}$$_{,δ}$ + W$_{2}$$_{,δ}$ ) M( U$_{i,δ}$ ) ≤ (11 L ) p − d $^{+1}$δ + (11 L ) p − d δ M( ∂U δ i ) M( ∂U δ i ) = (11 L ) p − $^{d}$(1 + 11 L ) δ (3.22a) M( W$_{j,δ}$ ) ≤ (11 L ) p − $^{d}$δ + (11 L ) p − d − 1 δ M( ∂W δ $_{j}$) M( ∂W δ $_{j}$) = (11 L ) p − d − $^{1}$(1 + 11 L ) δ (3.22b) M( U$_{i,δ}$ ) ≤ (11 L ) p − d $^{+1}$M( U δ i ) + (11 L ) p − $^{d}$Δ M( ∂U δ i ) , (3.21a) M( W$_{j,δ}$ ) ≤ (11 L ) p − $^{d}$M( W δ $_{j}$) + (11 L ) p − d − $^{1}$Δ M( ∂W δ $_{j}$)) . (3.21b) P$_{δ}$ = T + U δ 0 + ∂W δ 0 = X + ∂S + U δ 0 + ∂W δ 0 = X$_{δ}$ + U δ 0 + U δ 1 + ∂ ( S$_{δ}$ + W δ 0 + W δ 1 + W δ $_{2}$) . (3.20)