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 Th6:
  for t1,t2 being finite DecoratedTree st t1 in Subtrees t2
  holds height dom t1 <= height dom t2
  proof
    let t1,t2 be finite DecoratedTree; assume
    t1 in Subtrees t2;
    then consider p being Element of dom t2 such that
A1: t1 = t2|p;
    consider r being FinSequence of NAT such that
A2: r in dom t1 & len r = height dom t1 by TREES_1:def 12;
    dom t1 = (dom t2)|p by A1,TREES_2:def 10;
    then p^r in dom t2 by A2,TREES_1:def 6;
    then len (p^r) <= height dom t2 by TREES_1:def 12;
    then len p + len r <= height dom t2 & len r <= len p+len r
    by NAT_1:11,FINSEQ_1:22;
    hence height dom t1 <= height dom t2 by A2,XXREAL_0:2;
  end;
