reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th47:
  p in doms F iff len p = len F &
         for i st i in dom p holds p.i in dom (F.i)
proof
  hereby assume p in doms F;
    then ex q be FinSequence st q=p & len q = len F &
    for i st i in dom q holds q.i in dom (F.i) by Def8;
    hence len p = len F & for i st i in dom p holds p.i in dom (F.i);
  end;
  thus thesis by Def8;
end;
