If T is integral, then P can be taken to be integral as well. Proof. This is a slight modification of Federer’s theorems which do not state Equa- tions (3.10b) and (3.10c) separately but rather a combined bound M( P ) + M( ∂P ) ≤ M( T ) + M( ∂T ) + ρ . We show only the derivation of the separated bounds. In the normal current case [19, 4.2.24], these bounds follow from Federer’s proof. In particular, we have currents P$_{1}$ , P$_{2}$ and Y such that P = P$_{1}$ + Y and the following bounds hold: The bounds in Equations (3.10b) and (3.10c) follow from the triangle inequality and Equations (3.11a) to (3.11c): In the integral current case [19, 4.2.21], Federer applies the approximation theorem 4.2.20 to obtain P close to the pushforward of T under a Lipschitz diffeomorphism f . M( P ) ≤ M( P$_{1}$ ) + M( Y ) < M( T ) + ρ/ 4 + ρ/ 2 , M( ∂P ) = M( ∂P$_{1}$ + ∂Y ) ≤ M( P$_{2}$ − ∂P$_{1}$ − ∂Y ) + M( P$_{2}$ ) < ρ/ 2 + M( ∂T ) + ρ/ 4 . M( P$_{1}$ ) < M( T ) + ρ/ 4 , (3.11a) M( P$_{2}$ ) < M( ∂T ) + ρ/ 4 , (3.11b) M( P$_{2}$ − ∂P$_{1}$ − ∂Y ) + M( Y ) < ρ/ 2 . (3.11c) 62