Maxwell Equations in Thermodynamics lecture no. 5

we have four maxwell equations in thermodynamics which we have derived in this lecture:
In thermodynamics, the Maxwell equations are a set of equations derived by application of Euler's reciprocity relation to the thermodynamic characteristic functions. The Maxwell relations, first derived by James Clerk Maxwell, are the following expressions between partial differential quotients:
The characteristic functions are: U (internal energy), A (Helmholtz free energy), H (enthalpy), and G (Gibbs free energy). The thermodynamic parameters are: T (temperature), S (entropy), P (pressure), and V (volume).
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Starting with the Maxwell’s thermodynamic equation    =   show that there is a change in melting or boiling point with increase in pressure at constant temperature hence explain why water enclosed in a tightly covered container may be heated beyond its boiling point without evaporation

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How you get equal sine in between the partial derivatives of the equation 6,7,8,9.?

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