Proof. Unless otherwise proven, the data registered on the Application and/or the Website form the proof of the purchase of any transportation ticket or any booking by the User.
Proof. The information contained in the Bank’s records shall be deemed to constitute conclusive proof in the relationship between the Bank and the Account Holder, unless the Account Holder provides proof to the contrary. The Bank is not required to keep its records for a period longer than the statutory record keeping period.
Proof. Conjecture that a steady-state exists. Start by substituting (G.3)-(G.5) in (G.1) and setting the right hand side of (G.1) to 0. This gives the following quadratic equation:
Proof of Proposition 3 (V1 ≥ VO). The first step of the proof is provided by the following Lemma: Lemma G.2 If SB l∗(1, p), ρ∗(1, p) > 0, then SB l∗(8, p), ρ∗(8, p) ≥ 0 68 ∈ [0, 1] Proof. The case for 8 = 0 is trivial since the SB equals 0 if the lender has all the bargaining power. Define S(8) as short-hand for S(lj∗(8, p), ρj∗(8, p)). We prove the Lemma by contradic- tion. Suppose Iˆ8 ∈ [0, 1] such that SB(ˆ8) < 0. Since SB(8) is continuous in 8, I˜8 ∈ (ˆ8, 1) such that SB(˜8) = 0. This implies that SB(ˆ8)^8SL(ˆ8)1—^8 ≥ SB(˜8)^8SL(˜8)1—^8 = 0 ⇒ SL(ˆ8) = 0. If the lender’s surplus is equal to 0, this is only possible if lj∗(ˆ8, p)ρj∗(ˆ8, p) = 0. Since the loan contract is risk free, this implies that lj∗(ˆ8, p) = rj , resulting in SB(ˆ8) = SB(1) > 0, a
Proof. Applying the implicit function theorem to (G.39), the derivative is equal to the following expression:
Proof. We use a variant of Xxxxxx’x Existence and Uniqueness Theorem for ODEs to prove this proposition.2 The initial condition for the ODE is provided by p∗(0) Ξ pO, defined by (G.40).