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 Th16:
  for PA,PB being a_partition of Y holds PA '<' PA '\/' PB
proof
  thus PA '<' PA '\/' PB
  proof
 for a being set st a in PA ex b being set st b in PA '\/' PB & a c= b
    proof
      let a be set;
      assume
A1:   a in PA;
then A2:   a<>{} by EQREL_1:def 4;
      set x = the Element of a;
A3:   x in Y by A1,A2,TARSKI:def 3;
   union (PA '\/' PB) = Y by EQREL_1:def 4;
      then consider b being set such that
A4:   x in b and
A5:   b in PA '\/' PB by A3,TARSKI:def 4;
   b is_min_depend PA,PB by A5,Def5;
then    b is_a_dependent_set_of PA;
      then consider B being set such that
A6:  B c= PA and B<>{} and
A7:  b = union B;
  a in B
      proof
        consider u being set such that
A8:    x in u and
A9:    u in B by A4,A7,TARSKI:def 4;
    a /\ u <> {} by A2,A8,XBOOLE_0:def 4;
then A10:    not a misses u by XBOOLE_0:def 7;
    u in PA by A6,A9;
        hence thesis by A1,A9,A10,EQREL_1:def 4;
      end;
      hence thesis by A5,A7,ZFMISC_1:74;
    end;
    hence thesis by SETFAM_1:def 2;
  end;
end;
