Symmetry of net substitution effects is one of the principal conclusions of the theory of utility maximization.Which two mathematical theorems are used to prove this symmetry?

Question

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The Envelope Theorem and Young's Theorem are used to prove symmetry of net substitution effects, as they show that the substitution effect and income effect of a price change have the same magnitude but opposite signs. Taylor's Theorem and Lagrangian Theorem are not relevant to this concept, and Cauchy's Theorem and DeMoivre's Theorem are not applicable to utility maximization.

Symmetry of Net Substitution Effects Is One of the Principal