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 Th5:
  for t being Vertex of X-CircuitStr holds
  t in the carrier' of X-CircuitStr iff t is CompoundTerm of S,V
proof
  let t being Vertex of X-CircuitStr;
  thus t in the carrier' of X-CircuitStr implies t is CompoundTerm of S,V
  by Th4;
  set C = [:the carrier' of S, {the carrier of S}:];
  consider tt being Element of X, p being Node of tt such that
A1: t = tt|p by TREES_9:19;
  assume t is CompoundTerm of S,V;
  then reconsider t9 = t as CompoundTerm of S,V;
A2: t9.{} in C by MSATERM:def 6;
A3: p^(<*>NAT) = p by FINSEQ_1:34;
  {} in (dom tt)|p by TREES_1:22;
  then tt.p in C by A1,A2,A3,TREES_2:def 10;
  hence thesis by A1,TREES_9:24;
end;
