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, t being Term of S,X st t is not root
  ex o being OperSymbol of S st t.{} = [o,the carrier of S]
proof
  let S be non void non empty ManySortedSign, X be non-empty ManySortedSet
  of the carrier of S, t be Term of S,X;
  assume
A1: t is not root;
  per cases by MSATERM:2;
  suppose
    ex s being SortSymbol of S,v being Element of X.s st t.{} = [v,s];
    then consider s being SortSymbol of S, v being Element of X.s such that
A2: t.{} = [v,s];
    t = root-tree [v,s] by A2,MSATERM:5;
    hence thesis by A1;
  end;
  suppose
    t.{} in [:the carrier' of S,{the carrier of S}:];
    then consider o, c being object such that
A3: o in the carrier' of S and
A4: c in {the carrier of S} & t.{} = [o,c] by ZFMISC_1:def 2;
    reconsider o as OperSymbol of S by A3;
    take o;
    thus thesis by A4,TARSKI:def 1;
  end;
end;
