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 Th3:
  for PA,PB being a_partition of Y st PA '>' PB & PB '>' PA holds PB c= PA
proof
  let PA,PB be a_partition of Y;
  assume that
A1: PA '>' PB and
A2: PB '>' PA;
  let x be object;
   reconsider xx=x as set by TARSKI:1;
  assume
A3: x in PB;
  then consider V such that
A4: V in PA and
A5: xx c= V by A1,SETFAM_1:def 2;
  consider W being set such that
A6: W in PB and
A7: V c= W by A2,A4,SETFAM_1:def 2;
 x=W by A3,A5,A6,A7,Th1,XBOOLE_1:1;
  hence thesis by A4,A5,A7,XBOOLE_0:def 10;
end;
