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When both agents contribute we can substitute in the linear contribution schedules to get:
:<math>u_e(x^*) + u_g^*(x^*) + \tau_g + u_h^*(x^*) + \tau_h \ge u_e(x) + u_g^*(x) + \tau_g + u_h^*(x) + \tau_h \; \forall x \in \mathbb{R}</math>
:<math>\therefore u_e(x^*) + u_g^*(x^*) + u_h^*(x^*) \ge u_e(x) + u_g^*(x) + u_h^*(x) \; \forall x \in \mathbb{R}</math>
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