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
  for F,x holds
    doms (F^<* <*x*> *>) = {f^<* 1 *> where f is Element of doms F:f in doms F}
proof
  let F,x;
  1<= len F+1 & len (F^<* {} *>)=len F+1 by FINSEQ_2:16,NAT_1:11;
  then 1+len F in dom (F^<*{}*>) & (F^<*{}*>).(1+len F)={} by FINSEQ_3:25;
  then not (F^<*{}*>) is non-empty;
  then
A1: doms (F^<* {} *>) ={} by Th45;
  reconsider g={} as FinSequence;
  doms (F^<* g^<*x*> *>) = doms (F^<* g *>) \/
  {f^<*1 + len g*> where f is Element of doms F:f in doms F} by Th52;
  hence thesis by A1;
end;
