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 Th22:
  f is Tree-yielding iff for x st x in dom f holds f.x is Tree
proof
  thus f is Tree-yielding implies for x st x in dom f holds f.x is Tree
  proof
    assume
A1: for x st x in rng f holds x is Tree;
    let x;
    assume x in dom f;
    then f.x in rng f by FUNCT_1:def 3;
    hence thesis by A1;
  end;
  assume
A2: for x st x in dom f holds f.x is Tree;
  let x;
  assume x in rng f;
  then ex y being object st y in dom f & x = f.y by FUNCT_1:def 3;
  hence thesis by A2;
end;
