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 Th9:
  for f being SetSequence of Omega for X being Subset of Omega
  holds X/\ Union f= Union seqIntersection(X,f)
proof
  let f be SetSequence of Omega;
  let X be Subset of Omega;
A1: dom f=NAT by FUNCT_2:def 1;
  now
    reconsider g=seqIntersection(X,f) as SetSequence of Omega;
    let z be object;
    assume z in Union seqIntersection(X,f);
    then consider u such that
A2: z in u and
A3: u in rng g by TARSKI:def 4;
    consider v being object such that
A4: v in dom g and
A5: u=g.v by A3,FUNCT_1:def 3;
    reconsider n=v as Element of NAT by A4,FUNCT_2:def 1;
A6: z in X/\ f.n by A2,A5,Def1;
    then
A7: z in X by XBOOLE_0:def 4;
A8: f.n in rng f by A1,FUNCT_1:def 3;
    z in f.n by A6,XBOOLE_0:def 4;
    then z in Union f by A8,TARSKI:def 4;
    hence z in X/\ Union f by A7,XBOOLE_0:def 4;
  end;
  then
A9: Union seqIntersection(X,f) c= X/\ Union f;
A10: dom seqIntersection(X,f)=NAT by FUNCT_2:def 1;
  now
    let z be object;
    assume
A11: z in X/\ Union f;
    then z in union rng f by XBOOLE_0:def 4;
    then consider u such that
A12: z in u and
A13: u in rng f by TARSKI:def 4;
    consider v being object such that
A14: v in dom f and
A15: u=f.v by A13,FUNCT_1:def 3;
    reconsider n=v as Element of NAT by A14,FUNCT_2:def 1;
    X/\ f.n = (seqIntersection(X,f)).n by Def1;
    then
A16: X/\ f.n in rng seqIntersection(X,f) by A10,FUNCT_1:def 3;
    z in X by A11,XBOOLE_0:def 4;
    then z in X/\ f.n by A12,A15,XBOOLE_0:def 4;
    hence z in Union seqIntersection(X,f) by A16,TARSKI:def 4;
  end;
  then X/\ Union f c= Union seqIntersection(X,f);
  hence thesis by A9,XBOOLE_0:def 10;
end;
