I don't have much time tonight but note that going from the second equation to the last

 

[math]T = {\left( {\frac{{\partial U}}{{\partial S}}} \right)_{V,N}}[/math]

and

 

[math]P = - {\left( {\frac{{\partial U}}{{\partial V}}} \right)_{S,N}}[/math]

 

so the first term is

 

[math]S\frac{\partial }{{\partial \left( {\lambda S} \right)}}U\left( {\lambda S,\lambda V,\lambda N} \right)[/math]

 

This is therefore ST

 

similarly the second term leads to -PV

 

I will try to do more tomorrow.