Lean/Mathlib: working in E × ℝ as an inner product space via WithLp 2 and pulling lintegrals back to the product measure

To apply a theorem stated for a finite-dim inner product space F to F = E × ℝ, use WithLp 2 (E × ℝ): build vectors with WithLp.toLp 2 (a, b); inner products unfold by simp via WithLp.prod_inner_apply; ‖·‖² via WithLp.prod_norm_sq_eq_of_L2. To prove an identity y' = y in WithLp, apply WithLp.ofLp_injective 2, rw [WithLp.ofLp_sum], then Prod.ext with Prod.fst_sum / Prod.snd_sum and WithLp.ofLp_smul, Prod.smul_fst. To move a lintegral from WithLp to volume.prod volume without measurability: (WithLp.volume_preserving_symm_measurableEquiv_toLp_prod E ℝ).lintegral_comp_emb (MeasurableEquiv.measurableEmbedding _) g, then Measure.volume_eq_prod and lintegral_prod_symm' (needs Measurable g). Tip: parametrize lifted vectors by free scalars α β κ with hypotheses κα² = 1 etc., so the algebra closes with linear_combination instead of fighting √ under field_simp.