reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;
reserve C for Chain of W,
  B for Branch of W;
reserve T,T1,T2 for DecoratedTree;
reserve x1,x2 for set,
  w for FinSequence of NAT;

theorem
  for n being Nat holds TrivialInfiniteTree-level n = { n |-> 0 }
proof
  set T = TrivialInfiniteTree;
  let n be Nat;
  set L = { w where w is Element of T: len w = n };
  set f = n |-> 0;
  {f} = L
  proof
    hereby
      let a be object;
      assume a in {f};
      then
A1:   a = f by TARSKI:def 1;
      f in T & len f = n by CARD_1:def 7;
      hence a in L by A1;
    end;
    let a be object;
    assume a in L;
    then consider w being Element of T such that
A2: a = w & len w = n;
    w in T;
    then ex k being Nat st w = k |-> 0;
    then a = f by A2,CARD_1:def 7;
    hence thesis by TARSKI:def 1;
  end;
  hence thesis;
end;
