reserve IIG for Circuit-like non void non empty ManySortedSign;
reserve IIG for monotonic Circuit-like non void non empty ManySortedSign;

theorem Th9:
  for IIG for A being non-empty Circuit of IIG, v being Vertex of
  IIG, e being Element of (the Sorts of FreeEnv A).v st e.{} = [action_at v,the
  carrier of IIG] ex p being DTree-yielding FinSequence st e = [action_at v,the
  carrier of IIG]-tree p
proof
  let IIG;
  let A be non-empty Circuit of IIG, v be Vertex of IIG, e be Element of (the
  Sorts of FreeEnv A).v such that
A1: e.{} = [action_at v,the carrier of IIG];
  set X = the Sorts of A;
  NonTerminals(DTConMSA(X)) = [:the carrier' of IIG,{the carrier of IIG}:]
  & the carrier of IIG in {the carrier of IIG} by MSAFREE:6,TARSKI:def 1;
  then reconsider
  nt = [action_at v,the carrier of IIG] as NonTerminal of DTConMSA(
  X) by ZFMISC_1:87;
  FreeMSA(X) = MSAlgebra (# FreeSort(X), FreeOper(X) #) & e in (the Sorts
  of FreeEnv A).v by MSAFREE:def 14;
  then e in FreeSort(X,v) by MSAFREE:def 11;
  then reconsider tsg = e as Element of TS DTConMSA(X);
  consider ts being FinSequence of TS DTConMSA(X) such that
A2: tsg = nt-tree ts and
  nt ==> roots ts by A1,DTCONSTR:10;
  take ts;
  thus thesis by A2;
end;
