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,PB being a_partition of Y holds PA '/\' (PA '\/' PB) = PA
proof
  let PA,PB be a_partition of Y;
  ERl (PA '/\' (PA '\/' PB)) = ERl(PA) /\ ERl(PA '\/' PB) &
  ERl(PA) /\ ERl(PA '\/' PB) = ERl(PA) /\ (ERl(PA) "\/" ERl(PB))
    by Th23,Th24;
  hence thesis by Th25,EQREL_1:16;
end;
