reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;

theorem Th2:
  Subtrees root-tree x = {root-tree x}
  proof
A1: dom root-tree x = {{}} by TREES_4:3,TREES_1:29;
    thus Subtrees root-tree x c= {root-tree x}
    proof
      let y be object; assume y in Subtrees root-tree x; then
      consider p being Element of dom root-tree x such that
A2:   y = (root-tree x)|p;
      p = {} by A1,TARSKI:def 1;
      then y = root-tree x by A2,TREES_9:1;
      hence thesis by TARSKI:def 1;
    end;
    reconsider p = {} as Element of dom root-tree x by A1,TARSKI:def 1;
    let y be object; assume y in {root-tree x};
    then y = root-tree x by TARSKI:def 1;
    then y = (root-tree x)|p by TREES_9:1;
    hence thesis;
  end;
