The high energy gamma-rays produced by the ICS process e $^{±}$γ → e ± $^{′}$γ ′ have the following energy spectrum [37, 38]: Since the photon may be emitted from both electron and positron, an overall factor of 2 has been multiplied in the equation. The differential energy density u$_{γ}$ ( E$_{γ}$, r, z ) of interstellar radiation field (ISRF) contains three components: the cosmic microwave background (CMB), thermal dust radiation and star light. Here we adopt the GALPROP numerical result for u$_{γ}$ in Ref. [39]. The parameter f$_{ICS}$ is defined by [37] with where 0 ≤ q ≤ 1. In Eq. (35), f$_{e}$ + ( E$_{e}$, r, z ) is the positron differential number density. Because of the ICS and synchrotron, high energy electrons and positrons will loss most of their energy within a kpc. Therefore one may neglect the diffusion term of Eq. (16) and approximately calculate f$_{e}$ + ( E$_{e}$, r, z ) for every point in the CR propagation region. In this case, f$_{e}$ + ( E$_{e}$, r, z ) is given by the following formulas [37, 40] where the electron energy loss rate b$_{ICS}$ ( E$_{e}$, r, z ) is Here we neglect the synchrotron energy loss rate b$_{syn}$ as b$_{syn}$ ≪ b$_{ICS}$ [38]. It is worthwhile to stress that the above energy loss rate b$_{ICS}$ ( E$_{e}$, r, z ) is position dependent $^{1}$. In order to compare with the experimental data, one needs to know the diffuse gamma- ray background which includes a galactic Φ Galactic γ contribution and an extragalactic (EG) Φ EG γ contribution. The galactic gamma-ray background Φ Galactic γ mainly comes from pion 1 If a position independent energy loss rate is assumed, one can use Eq. (19) to calculate f$_{e}$ + ( E$_{e}$, r, z ). 19 b$_{ICS}$ ( E$_{e}$, r, z ) = 2 πα 2 em E 2 e ∫ dE$_{γ}$ u$_{γ}$ ( E$_{γ}$, r, z ) E 2 γ ∫ dE$_{γ}$ ′ ( E$_{γ}$ ′ − E$_{γ}$ ) f$_{ICS}$ . (39) f$_{e}$ + ( E$_{e}$, r, z ) = 1 b$_{ICS}$ ( E$_{e}$, r, z ) ρ ( r, z ) m$_{D}$ ∑ k Γ$_{k}$ ∫ m$_{D}$ E$_{e}$ dE $_{′}$dn k e + dE ′ , (38) ϵ = E$_{γ}$ ′ E$_{e}$ , q = E$_{γ}$ ′ m 2 e 4 E$_{γ}$E$_{e}$ ( E$_{e}$ − E$_{γ}$ ′ ) , (37) f$_{ICS}$ = 2 q ln q + (1 + 2 q )(1 − q ) + 1 2 ϵ 2 1 − ϵ (1 − q ) , (36) d Φ$_{γ}$ ′ dE$_{γ}$ ′ = α 2 em ΔΩ ∫ ΔΩ d Ω$^{∫}$ LOS ds$^{∫}$ ∫ f$_{e}$ + ( E$_{e}$, r, z ) u$_{γ}$ ( E$_{γ}$, r, z ) f$_{ICS}$ dE$_{e}$ E 2 e dE$_{γ}$ E 2 γ . (35)