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;
A1: PA '<' PA '\/' PA by Th16;
 for a being set st a in PA '\/' PA ex b being set st b in PA & a c= b
  proof
    let a be set;
    assume
A2: a in PA '\/' PA; then
A3: a is_min_depend PA,PA by Def5;
    then a is_a_dependent_set_of PA;
    then consider B being set such that
A4: B c= PA and B<>{} and
A5: a = union B;
A6: a <> {} by A2,EQREL_1:def 4;
    set x = the Element of a;
 x in a by A6;
    then x in Y by A2;
    then x in union PA by EQREL_1:def 4;
    then consider b being set such that
A7: x in b and
A8: b in PA by TARSKI:def 4;
    b in B
    proof
      consider u being set such that
A9:  x in u and
A10:  u in B by A5,A6,TARSKI:def 4;
  b /\ u <> {} by A7,A9,XBOOLE_0:def 4;
then A11:  not b misses u by XBOOLE_0:def 7;
  u in PA by A4,A10;
      hence thesis by A8,A10,A11,EQREL_1:def 4;
    end;
then A12: b c= a by A5,ZFMISC_1:74;
 b is_a_dependent_set_of PA by A8,Th6;
then  a = b by A3,A12;
    hence thesis by A8;
  end;
then  PA '\/' PA '<' PA by SETFAM_1:def 2;
  hence thesis by A1,Th4;
end;
