  For background gauge field configurations reducible to the form Amu = (A3,
A(x)) where A3 is a constant, we provide an elementary derivation of the
recently obtained result for the exact induced Chern-Simons (CS) effective
action in QED3 at finite temperature. The method allows us to extend the result
in several useful ways: to obtain the analogous result for the `mixed' CS term
in the Dorey-Mavromatos model of parity-conserving planar superconductivity,
thereby justifying their argument for flux quantization in the model; to the
induced CS term for a tau-dependent flux; and to the term of second order in
A(x) (and all orders in A3) in the effective action.
