reserve G for Graph,
  k, m, n for Nat;
reserve G for non void Graph;

theorem
  for S being non void non empty ManySortedSign, X being non-empty
  ManySortedSet of the carrier of S, v, vk being SortSymbol of S, o being
  OperSymbol of S, t being Element of (the Sorts of FreeMSA X).v, a being (
  ArgumentSeq of Sym(o,X)), k being Element of NAT, ak being Element of (the
Sorts of FreeMSA X).vk st t = [o,the carrier of S]-tree a & k in dom a & ak = a
  .k holds depth ak < depth t
proof
  let S be non void non empty ManySortedSign, X be non-empty ManySortedSet of
the carrier of S, v, vk be SortSymbol of S, o be OperSymbol of S, t be Element
of (the Sorts of FreeMSA X).v, a be (ArgumentSeq of Sym(o,X)), k be Element of
  NAT, ak be Element of (the Sorts of FreeMSA X).vk;
  assume that
A1: t = [o,the carrier of S]-tree a and
A2: k in dom a and
A3: ak = a.k;
  reconsider a9 = a as DTree-yielding FinSequence;
A4: (ex dt being finite DecoratedTree, tt being finite Tree st dt = t & tt =
dom dt & depth t = height tt )& ex q being DTree-yielding FinSequence st a9 = q
  & dom t = tree doms q by A1,MSAFREE2:def 14,TREES_4:def 4;
  reconsider da = doms a as FinTree-yielding FinSequence;
  consider dtk being finite DecoratedTree, ttk being finite Tree such that
A5: dtk = ak & ttk = dom dtk and
A6: depth ak = height ttk by MSAFREE2:def 14;
  dom doms a9 = dom a9 & ttk = da.k by A2,A3,A5,FUNCT_6:22,TREES_3:37;
  then ttk in rng da by A2,FUNCT_1:def 3;
  hence thesis by A6,A4,TREES_3:78;
end;
