Is the expression x space logical or space (y space logical and space z )equivalent to (x space logical or space y )space logical and space (x space logical or space z )for all possible inputs of x, y, and z

To prove:

X+Y.Z=(X+Y).(X+Z)

Taking R.H.S

= (X+Y).(X+Z)

By distributive law

= X.X+X.Z+X.Y+Y.Z --- (1)

From Boolean algebra

X.X = X

X.Y+X.Z = X.(Y+Z)

Using these in (1)

=X+X(Y+Z)+Y.Z

=X(1+(Y+Z)+Y.Z --- (2)

As we know (1+X) = 1

Then (2) becomes

=X.1+Y.Z

=X+Y.Z

Which is equal to R.H.S

Hence proved,

X+Y.Z=(X+Y).(X+Z)