Base-independence of Fréchet differentiability without conditions (SX) and (SY)

Prove that Theorem 4.23 holds without the structural assumptions (SX) on the domain space X and (SY) on the range space Y; specifically, show that if two different families of positive F-seminorms independently induce the same topologies on X and Y, then Fréchet differentiability at a point and the value of the derivative are independent of the chosen family of F-seminorms.

Background

Theorem 4.23 proves that Fréchet differentiability and the derivative are invariant under changing between two families of F-seminorms that induce the same topologies on X and Y, provided additional comparability assumptions (SX) and (SY) hold.

The authors ask whether these assumptions can be dropped, which would establish that Fréchet differentiability is an intrinsic property of the topology, independent of the particular F-seminorm basis chosen.