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 Th5:
  for f being SetSequence of Omega holds disjointify(f) is disjoint_valued
proof
  let f be SetSequence of Omega;
  now
    let n,m;
    assume n<m;
    then
A1: n in Segm m by NAT_1:44;
    dom f=NAT by FUNCT_2:def 1;
    then dom(f|m)=m/\ NAT by RELAT_1:61;
    then n in dom(f|m) by A1,XBOOLE_0:def 4;
    then
A2: (f|m).n in rng(f|m) by FUNCT_1:def 3;
    (f|m).n=f.n by A1,FUNCT_1:49;
    then f.n misses f.m \ union rng(f|m) by A2,XBOOLE_1:85,ZFMISC_1:74;
    then
A3: f.n misses (disjointify(f)).m by Th4;
    f.n \ union rng(f|n) c= f.n by XBOOLE_1:36;
    then (disjointify(f)).n c= f.n by Th4;
    hence (disjointify(f)).n misses (disjointify(f)).m by A3,XBOOLE_1:63;
  end;
  hence thesis;
end;
