reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;

theorem Th34:
  i <> 0 implies (i |-> x is Tree-yielding iff x is Tree)
proof
  assume
A1: i <> 0;
  i |-> x = (Seg i) --> x by FINSEQ_2:def 2;
  then rng (i |-> x) = {x} by A1,FUNCOP_1:8;
  then x is Tree iff rng (i |-> x) is constituted-Trees by Th12;
  hence thesis;
end;
