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 Th9:
  for X0,X1 being Subset of Y st X0 is_a_dependent_set_of PA &
  X1 is_a_dependent_set_of PA & X0 meets X1
  holds X0 /\ X1 is_a_dependent_set_of PA
proof
  let X0,X1 be Subset of Y;
  assume that
A1: X0 is_a_dependent_set_of PA and
A2: X1 is_a_dependent_set_of PA and
A3: X0 meets X1;
  consider A being set such that
A4: A c= PA and A<>{} and
A5: X0 = union A by A1;
  consider B being set such that
A6: B c= PA and B<>{} and
A7: X1 = union B by A2;
A8: X0 /\ X1 c= union (A /\ B)
  proof
    let x be object;
    assume
A9: x in (X0 /\ X1);
then A10: x in X0 by XBOOLE_0:def 4;
A11: x in X1 by A9,XBOOLE_0:def 4;
    consider y being set such that
A12: x in y and
A13: y in A by A5,A10,TARSKI:def 4;
    consider z being set such that
A14: x in z and
A15: z in B by A7,A11,TARSKI:def 4;
A16: y in PA by A4,A13;
A17: z in PA by A6,A15;
 y meets z by A12,A14,XBOOLE_0:3;
    then y = z by A16,A17,EQREL_1:def 4;
    then y in A /\ B by A13,A15,XBOOLE_0:def 4;
    hence thesis by A12,TARSKI:def 4;
  end;
  union (A /\ B) c= X0 /\ X1
  proof
    let x be object;
    assume x in union (A /\ B);
    then consider y being set such that
A18: x in y and
A19: y in (A /\ B) by TARSKI:def 4;
A20: y in A by A19,XBOOLE_0:def 4;
A21: y in B by A19,XBOOLE_0:def 4;
A22: x in X0 by A5,A18,A20,TARSKI:def 4;
 x in X1 by A7,A18,A21,TARSKI:def 4;
    hence thesis by A22,XBOOLE_0:def 4;
  end;
then A23: X0 /\ X1 = union (A /\ B) by A8,XBOOLE_0:def 10;
A24: A \/ B c= PA by A4,A6,XBOOLE_1:8;
 A /\ B c= A & A c= A \/ B by XBOOLE_1:7,17;
then  A /\ B c= A \/ B;
then A25: A /\ B c= PA by A24;
 now
    assume
A26: A /\ B={};
 ex y being object st y in X0 & y in X1 by A3,XBOOLE_0:3;
    hence contradiction by A8,A26,XBOOLE_0:def 4,ZFMISC_1:2;
  end;
  hence thesis by A23,A25;
end;
