reserve Omega, F for non empty set,
  f for SetSequence of Omega,
  X,A,B for Subset of Omega,
  D for non empty Subset-Family of Omega,
  n,m for Element of NAT,
  h,x,y,z,u,v,Y,I for set;

theorem Th6:
  for f being SetSequence of Omega holds union rng disjointify(f) = union rng f
proof
  let f be SetSequence of Omega;
A1: dom f=NAT by FUNCT_2:def 1;
A2: dom disjointify f=NAT by FUNCT_2:def 1;
  now
    let x be object;
    defpred P[Nat] means x in f.$1;
    assume x in union rng f;
    then consider y such that
A3: x in y and
A4: y in rng f by TARSKI:def 4;
    consider z being object such that
A5: z in dom(f) and
A6: y=f.z by A4,FUNCT_1:def 3;
    reconsider n=z as Element of NAT by A5,FUNCT_2:def 1;
A7: ex k being Nat st P[k]
    proof
      take n;
      thus thesis by A3,A6;
    end;
    consider k being Nat such that
A8: P[k] & for m being Nat st P[m] holds k <= m from NAT_1:sch 5(A7);
    now
      assume x in union rng(f|k);
      then consider y such that
A9:   x in y and
A10:  y in rng(f|k) by TARSKI:def 4;
      consider z being object such that
A11:  z in dom(f|k) and
A12:  y=(f|k).z by A10,FUNCT_1:def 3;
      dom(f|k) c= NAT by A1,RELAT_1:60;
      then reconsider n=z as Element of NAT by A11;
      dom(f|k) c= Segm k by RELAT_1:58;
      then n<k & y=f.n by A11,A12,FUNCT_1:49,NAT_1:44;
      hence contradiction by A8,A9;
    end;
    then x in f.k \ union rng(f|k) by A8,XBOOLE_0:def 5;
    then
A13: x in (disjointify(f)).k by Th4;
    k in NAT by ORDINAL1:def 12;
    then (disjointify f).k in rng disjointify(f) by A2,FUNCT_1:def 3;
    hence x in union rng disjointify(f) by A13,TARSKI:def 4;
  end;
  then
A14: union rng f c= union rng disjointify(f);
  now
    let x be object;
    assume x in union rng disjointify(f);
    then consider y such that
A15: x in y and
A16: y in rng disjointify(f) by TARSKI:def 4;
    consider z being object such that
A17: z in dom(disjointify(f)) and
A18: y=(disjointify(f)).z by A16,FUNCT_1:def 3;
    reconsider n=z as Element of NAT by A17,FUNCT_2:def 1;
A19: f.n\ union rng (f|n) c= f.n & f.n in rng(f) by A1,FUNCT_1:def 3
,XBOOLE_1:36;
    x in f.n\ union rng (f|n) by A15,A18,Th4;
    hence x in union rng f by A19,TARSKI:def 4;
  end;
  then union rng disjointify(f) c= union rng f;
  hence thesis by A14,XBOOLE_0:def 10;
end;
