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 Th6:
  for PA,PB being a_partition of Y
  st PA '>' PB holds for b being set st b in PA holds
  b is_a_dependent_set_of PB
proof
  let PA,PB be a_partition of Y;
  assume
A1: PA '>' PB;
A2: union PB = Y by EQREL_1:def 4;
A3: PA is_coarser_than PB by A1,Th5;
 for b being set st b in PA holds b is_a_dependent_set_of PB
  proof
    let b be set;
    assume
A4: b in PA;
    set B0={x8 where x8 is Subset of Y: x8 in PB & x8 c= b};
    consider xb be set such that
A5: xb in PB & xb c= b by A3,A4,SETFAM_1:def 3;
A6: xb in B0 by A5;
   for z being object holds z in B0 implies z in PB
    proof let z be object;
      assume z in B0;
then   ex x8 being Subset of Y st x8=z & x8 in PB & x8 c= b;
      hence thesis;
    end;
then A7: B0 c= PB;
  for z being object holds z in b implies z in union B0
    proof let z be object;
      assume
A8:  z in b;
      then consider x1 such that
A9:  z in x1 and
A10:  x1 in PB by A2,A4,TARSKI:def 4;
      consider y1 such that
A11:  y1 in PA and
A12:  x1 c= y1 by A1,A10,SETFAM_1:def 2;
      b = y1 or b misses y1 by A4,A11,EQREL_1:def 4;
      then x1 in B0 by A8,A9,A10,A12,XBOOLE_0:3;
      hence thesis by A9,TARSKI:def 4;
    end; then
    A13: b c= union B0;
   for z being object holds z in union B0 implies z in b
    proof let z be object;
      assume z in union B0;
      then consider y such that
A14:  z in y and
A15:  y in B0 by TARSKI:def 4;
  ex x8 being Subset of Y st x8=y & x8 in PB & x8 c= b by A15;
      hence thesis by A14;
    end;
    then union B0 c= b;
    then b = union B0 by A13,XBOOLE_0:def 10;
    hence thesis by A6,A7;
  end;
  hence thesis;
end;
