Sequent Calculus Formula Doubt for a->b at left position

Question

According to the rules of sequent calculus, a->b on left of is

{} |- a   {} |- b  

 but in this example different rule is applied. Why is that the case?

Answer 1

I think that the same rule is applied: There is the implication c->b|d, so let us abbreviate A:=c and B->b|d, then the sequent A->B,Gamma |- Delta branches into Gamma |- A,Delta and B,Gamma |- Delta, right?