Prove G is a group

There exists an element e in G such that e*x=x for all x in G…(1)
Given x in G,there exists a y in G such that y*x=e…(2)
Given y in G,there exists a z in G such that z*y=e…(3)
First x*y=e*(x*y);from (1)
=(z*y)*(x*y);from (3)
=z*(y*x)*y;associative property
=z*e*y
=z*(e*y);associative property
=z*y
=e ; from(3)
Second, x*e = x*(y*x) from (2)
=(x*y)*x; associative property
=e*x; from above
=x; from (1)