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 Th5:
  for PA,PB being a_partition of Y st PA '>' PB holds PA is_coarser_than PB
proof
  let PA,PB be a_partition of Y;
  assume
A1: PA '>' PB;
 for x being set st x in PA ex y being set st y in PB & y c= x
  proof
    let x be set;
    assume
A2: x in PA;
then A3: x<>{} by EQREL_1:def 4;
    set u = the Element of x;
A4: u in x by A3;
    union PB = Y by EQREL_1:def 4;
    then consider y being set such that
A5: u in y and
A6: y in PB by A2,A4,TARSKI:def 4;
    consider z being set such that
A7: z in PA and
A8: y c= z by A1,A6,SETFAM_1:def 2;
    x=z or x misses z by A2,A7,EQREL_1:def 4;
    hence thesis by A3,A5,A6,A8,XBOOLE_0:3;
  end;
  hence thesis by SETFAM_1:def 3;
end;
