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 Th2:
  union (SFX \ {{}}) = union SFX
proof
A1: union (SFX \ {{}}) c= union SFX
  proof
    let y be object;
    assume y in union (SFX \ {{}});
    then ex z being set st y in z & z in (SFX \{{} }) by TARSKI:def 4;
    hence thesis by TARSKI:def 4;
  end;
  union SFX c= union (SFX \ {{}})
  proof
    let y be object;
    assume y in union SFX;
    then consider X being set such that
A2: y in X and
A3: X in SFX by TARSKI:def 4;
 not X in {{}} by A2,TARSKI:def 1;
then  X in SFX \ {{}} by A3,XBOOLE_0:def 5;
    hence thesis by A2,TARSKI:def 4;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
