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 Th10:
  for X being Subset of Y holds X is_a_dependent_set_of PA & X<>Y
  implies X` is_a_dependent_set_of PA
proof
  let X be Subset of Y;
  assume that
A1: X is_a_dependent_set_of PA and
A2: X<>Y;
  consider B being set such that
A3: B c= PA and B<>{} and
A4: X=union B by A1;
  take PA \ B;
A5: union PA = Y by EQREL_1:def 4;
then A6: X` = union PA \ union B by A4,SUBSET_1:def 4;
  reconsider B as Subset of PA by A3;
 now
    assume PA \ B={};
then  PA c= B by XBOOLE_1:37;
    hence contradiction by A2,A4,A5,XBOOLE_0:def 10;
  end;
  hence thesis by A6,EQREL_1:43,XBOOLE_1:36;
end;
