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 PA '/\' PA = PA
proof
  let PA be a_partition of Y;
 for u being set st u in INTERSECTION(PA,PA) \ {{}} ex v being set
  st v in PA & u c= v
  proof
    let u be set;
    assume u in INTERSECTION(PA,PA) \ {{}};
    then consider v,u2 being set such that
A1: v in PA and u2 in PA and
A2: u = v /\ u2 by SETFAM_1:def 5;
    take v;
    thus thesis by A1,A2,XBOOLE_1:17;
  end;
then A3: INTERSECTION(PA,PA) \ {{}} '<' PA by SETFAM_1:def 2;
 for u being set st u in PA
  ex v being set st v in INTERSECTION(PA,PA) \ {{}} & u c= v
  proof
    let u be set;
    assume
A4: u in PA;
then A5: u <> {} by EQREL_1:def 4;
    set v = u /\ u;
A6: not v in {{}} by A5,TARSKI:def 1;
 v in INTERSECTION(PA,PA) by A4,SETFAM_1:def 5;
then  v in INTERSECTION(PA,PA) \ {{}} by A6,XBOOLE_0:def 5;
    hence thesis;
  end;
then  PA '<' INTERSECTION(PA,PA) \ {{}} by SETFAM_1:def 2;
  hence thesis by A3,Th4;
end;
