reserve i,j,x,y for object,
  f,g for Function;
reserve T,T1 for finite Tree,
  t,p for Element of T,
  t1 for Element of T1;

theorem
  for q being DTree-yielding FinSequence, k being Nat st
    k+1 in dom q holds <*k*> in tree doms q
proof
  let q be DTree-yielding FinSequence, k be Nat;
A1: k < k+1 by XREAL_1:29;
A2: dom q = Seg len q & Seg len doms q = dom doms q by FINSEQ_1:def 3;
  assume
A3: k+1 in dom q;
  then k+1 <= len q by FINSEQ_3:25;
  then k < len q by A1,XXREAL_0:2;
  then
A4: k < len doms q by A2,FINSEQ_1:6,TREES_3:37;
  dom doms q = dom q by TREES_3:37;
  then (doms q).(k+1) is Tree by A3,TREES_3:22;
  then
A5: {} in (doms q).(k+1) by TREES_1:22;
  <*k*> = <*k*>^{} by FINSEQ_1:34;
  hence thesis by A5,A4,TREES_3:def 15;
end;
