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 Th8:
  for f being SetSequence of Omega holds f is disjoint_valued
  implies for X being Subset of Omega holds seqIntersection(X,f) is
  disjoint_valued
proof
  let f be SetSequence of Omega;
  assume
A1: f is disjoint_valued;
  let X be Subset of Omega;
  for n,m holds n<m implies (seqIntersection(X,f)).n misses (
  seqIntersection(X,f)).m
  proof
    let n,m;
    assume n<m;
    then f.n misses f.m by A1;
    then
A2: X/\ f.n misses f.m by XBOOLE_1:74;
    (seqIntersection(X,f)).n=X/\ f.n & (seqIntersection(X,f)).m=X/\ f.m by Def1
;
    hence thesis by A2,XBOOLE_1:74;
  end;
  hence thesis;
end;
