Prove that if a has a right inverse x and a left inverse y t

Prove that if a has a right inverse x and a left inverse y, then a is invertible, and its inverse is equal to x and to y.

Since a has a right inverse x

ax = e where e is the identity element

Similarly since a has left inverse y

ya =e

As closure property is true

a(xy)a = y(aa)x = e

Or Multiply by y one left and x on right

ya(xy) ax = yy(aa)(xx)

e(xy)e = (yy)(aa)(xx)

xy = y(ya) (ax)x = yeex = yx

i.e. xy = yx

Hence xya = yxa

Or x = yxa

Multiply by a inverse on right

x a^-1 = yx

Since xy = yx

we get a^-1=x

Similarly it can be proved that a^-1=y

Thus proved