reserve x for set,
  t,t1,t2 for DecoratedTree;
reserve C for set;
reserve X,Y for non empty constituted-DTrees set;
reserve T for DecoratedTree,
  p for FinSequence of NAT;
reserve T for finite-branching DecoratedTree,
  t for Element of dom T,
  x for FinSequence,
  n, m for Nat;
reserve x, x9 for Element of dom T,
  y9 for set;
reserve n,k1,k2,l,k,m for Nat,
  x,y for set;

theorem Th44:
  for T being Tree holds T-level 0 = {{}}
proof
  let T be Tree;
A1: {{}} c= { w where w is Element of T : len w = 0 }
  proof
    let x be object;
    assume x in {{}};
    then
A2: x = {} by TARSKI:def 1;
    {} in T by TREES_1:22;
    then consider t being Element of T such that
A3: t = {};
    len t = 0 by A3;
    hence thesis by A2,A3;
  end;
  { w where w is Element of T : len w = 0 } c= {{}}
  proof
    let x be object;
    assume x in { w where w is Element of T : len w = 0 };
    then consider w being Element of T such that
A4: w = x and
A5: len w = 0;
    w = {} by A5;
    hence thesis by A4,TARSKI:def 1;
  end;
  hence thesis by A1;
end;
