Prove or disprove: If f:X---->Yhas 1 or more left inverses g:Y---->X but possesses no right inverse, then f has more than one left inverse.

I am inclined to think it is true.

Assume f only has one such inverse, i.e. g is unique.

If f has no right inverse, there exists no map h such that f(h(a))=a for all a in X.

g(f(a))=a for all a in X.

stuck.