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 Th4:
  for PA,PB being a_partition of Y st PA '>' PB & PB '>' PA holds PA = PB
proof
  let PA,PB be a_partition of Y;
  assume PA '>' PB & PB '>' PA;
then  PB c= PA & PA c= PB by Th3;
  hence thesis by XBOOLE_0:def 10;
end;
