reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem
  for PA being a_partition of Y holds
  %O(Y) '\/' PA = %O(Y) & %O(Y) '/\' PA = PA
proof
  let PA be a_partition of Y;
A1: ERl(%O(Y) '\/' PA) = ERl(%O(Y)) "\/" ERl(PA) by Th23;
 ERl(%O(Y)) = nabla Y by Th33;
then  ERl(%O(Y)) \/ ERl(PA) = ERl(%O(Y)) by EQREL_1:1;
then  ERl(%O(Y)) c= ERl(%O(Y)) "\/" ERl(PA) by EQREL_1:def 2;
then A2: %O(Y) '<' %O(Y) '\/' PA by A1,Th20;
 %O(Y) '>' PA '\/' %O(Y) by Th32;
  hence %O(Y) '\/' PA = %O(Y) by A2,Th4;
   ERl(%O(Y) '/\' PA) = ERl(%O(Y)) /\ ERl(PA) & ERl(%O(Y)) = nabla Y by Th24
,Th33;
  hence %O(Y) '/\' PA = PA by Th25,XBOOLE_1:28;
end;
