[en] A bremsstrahlung amplitude in the special two-energy-two-angle (TETAS) approximation, which is relativistic, gauge invariant, and consistent with the soft-photon theorem, is derived for the pion-proton bremsstrahlung (π⁺pγ) process near the Δ⁺⁺(1232) resonance. In order to take into account bremsstrahlung emission from an internal Δ⁺⁺ line with both charge and the anomalous magnetic moment λ_Δ, we have applied a radiation decomposition identity to modify Low's standard prescription for constructing a soft-photon amplitude. This modified procedure is very general; it can be used to derive the TETAS amplitude for any bremsstrahlung process with resonance. The derived TETAS amplitude is applied to calculate all π⁺pγ cross sections which can be compared with the experimental data. Treating λ_Δ as a free parameter in these calculations, we extract the ''experimental'' magnetic moment of the Δ⁺⁺, μ_Δ, from recent data. The extracted values of μ_Δ are (3.7--4.2)e/(2m_p) from the University of California, Los Angeles data and (4.6--4.9)e/(2m_p) from the Paul Scherrer Institute data. Here, m_p is the proton mass