State and prove bernaullis theorem?

This theorem states that for stream line flow of an ideal liquid, the sum of energy (Pressure energy, potential energy and kinetic energy) per unit mass remain constant at every cross section through out the liquid flow. Consider an ideal liquid flow through a pipe of non uniform cross section. Let a1 and a2 = area of cross section at the end A and B respectively, v1 and v2 =velocity of liquid at the end of A and B respectively, p1 and p2= pressure at the end A and B respectively, h1 and h2 = height of end A and B from a reference level. Work done per second at the end A on the liquid = p1a1v1 Work done per second at the end B on the liquid = p2a2v2 From equation of continuity, we know a1v1¶ = a2v2¶ (¶ be the density of the liquid) Or, a1v1=a2v2=V (volume of the liquid flowing per second through the pipe) Net work done per second on liquid by pressure = p1a1v1 - p2a2v2 =p1V-p2V______1 Increase in kinetic energy of liquid per second from A to B =1/2mv2 square - 1/2mv1 square_____2 Increase in potential energy per second from A to B = mgh2-mgh1______3 Work done on liquid by pressure is the sum of increase in kinetic energy and potential energy. p1V-p2V= 1/2 mv2 square - 1/2 mv1 square + mgh2-mgh1 p1V+1/2 mv1 square + mgh1 = p2V + 1/2 mv2 square + mgh2 p1/¶ + 1/2 v1 square + gh1 = p2/¶ + 1/2 v2 square + gh2 p/¶ + 1/2 V square +gh = constant Multiplying above equation by "¶", P+1/2 ¶ V square + ¶gh = constant.