[Question] Is the sum individual informativeness of two independent variables no more than their joint informativeness?

Is it true that:

If I(X;Y) = 0 then I(S;X) + I(S;Y) ⇐ I(S;X,Y)

Can you find a coun­terex­am­ple, or prove this and teach me your proof?

Some­one showed me a sim­ple an­a­lytic proof. I am still in­ter­ested in see­ing differ­ent ways peo­ple might prove this though.