reserve S for non empty non void ManySortedSign,
  V for non-empty ManySortedSet of the carrier of S,
  A for non-empty MSAlgebra over S,
  X for non empty Subset of S-Terms V,
  t for Element of X;

theorem Th4:
  (for v being Vertex of X-CircuitStr holds v is Term of S,V) &
  for s being set st s in the carrier' of X-CircuitStr holds
  s is CompoundTerm of S,V
proof
  set C = [:the carrier' of S, {the carrier of S}:];
  hereby
    let v be Vertex of X-CircuitStr;
    consider t being Element of X, p being Node of t such that
A1: v = t|p by TREES_9:19;
    thus v is Term of S,V by A1;
  end;
  let s be set;
  assume s in the carrier' of X-CircuitStr;
  then consider t being Element of X, p being Node of t such that
A2: s = t|p and
A3: not p in Leaves dom t or t.p in C by TREES_9:24;
  reconsider s as Term of S,V by A2;
  reconsider e = {} as Node of t|p by TREES_1:22;
A4: dom (t|p) = (dom t qua Tree)|(p qua FinSequence of NAT) by TREES_2:def 10;
  p = p^e by FINSEQ_1:34;
  then
A5: t.p = s.e by A2,A4,TREES_2:def 10;
  p in Leaves dom t iff s is root by A2,TREES_9:6;
  hence thesis by A3,A5,MSATERM:28,def 6;
end;
