Chern data survive addition of trivial bundles

The finite-type $k$-scheme $X$ is noetherian and qcqs, so apply [F3] to $E,F$. Every integral cycle and every generating principal-divisor relation is supported in one of the open-and-closed strata, so the cycle groups and their rational-equivalence quotients split as finite direct sums over those strata. Thus Chow classes and polynomials are determined by their restrictions there, justifying the componentwise convention in the statement. The nonempty strata remain smooth equidimensional finite-type $k$-schemes, so [F1] and [F2] apply on each; for rank zero use the zero-bundle convention $c_t(0)=1$ in [F2].

On each stratum the $m=0$ equality is an identity. For $m>0$, Whitney applied $m$ times gives $c_t(E\oplus {\mathcal{O}}_X^m)=c_t(E)$, since each trivial line contributes the unit polynomial; the same holds for $F$. Step 1.1 assembles these equalities on $X$. If $E,F$ have rank two and equal Chern tuples, their first two coefficients agree by [F1] and both third coefficients vanish by [F2]. Stabilization therefore gives identical first three coefficients, with third coefficient zero.

Both schemes and $f$ have the specified base-field and finiteness scope. Pullback preserves direct sums and trivial bundles and pulls the rank partition back to an open-and-closed partition of $X^{\mathrm{\prime }}$. On each nonempty inverse-image stratum naturality in [F2] gives $c_i(f^{\ast }E)=f^{\ast }{c}_i(E)$ and the analogous identities for $F$ and their stabilizations. Apply this coefficientwise to step 2.1 and assemble over the strata as in step 1.1. All asserted equalities persist on $X^{\mathrm{\prime }}$, including the zero third coefficient.