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 Th3:
  X is SetWithCompoundTerm of S,V iff X-CircuitStr is non void
proof
  hereby
    assume X is SetWithCompoundTerm of S,V;
    then consider t being CompoundTerm of S, V such that
A1: t in X by MSATERM:def 7;
    t.{} in [:the carrier' of S, {the carrier of S}:] by MSATERM:def 6;
    hence X-CircuitStr is non void by A1,TREES_9:25;
  end;
  assume X-CircuitStr is non void;
  then consider t such that
A2: t is not root or t.{} in [:the carrier' of S, {the carrier of S}:]
  by TREES_9:25;
  t is CompoundTerm of S,V by A2,MSATERM:28,def 6;
  hence thesis by MSATERM:def 7;
end;
