
theorem Th125:
  for F being non empty Graph-yielding Function, S being GraphSum of F holds
    (F is acyclic iff S is acyclic) &
    (F is chordal iff S is chordal)
proof
  let F be non empty Graph-yielding Function, S be GraphSum of F;
  consider G9 being GraphUnion of rng canGFDistinction(F) such that
    A1: S is G9-Disomorphic by Def27;
  consider H being PGraphMapping of G9, S such that
    A2: H is Disomorphism by A1, GLIB_010:def 24;
  F, canGFDistinction(F) are_Disomorphic by Th87;
  then A3: F, canGFDistinction(F) are_isomorphic by Th42;
  :: acyclic
  thus F is acyclic implies S is acyclic by A2, GLIB_010:140;
  hereby
    assume S is acyclic;
    then G9 is acyclic by A2, GLIB_010:140;
    then rng canGFDistinction(F) is acyclic by Th60;
    hence F is acyclic by GLIB_014:3;
  end;
  :: chordal
  hereby
    assume F is chordal;
    then canGFDistinction(F) is chordal by A3, Th48;
    then G9 is chordal by Th63;
    hence S is chordal by A2, GLIB_010:140;
  end;
  hereby
    assume S is chordal;
    then G9 is chordal by A2, GLIB_010:140;
    then rng canGFDistinction(F) is chordal by Th63;
    then canGFDistinction(F) is chordal by GLIB_014:3;
    hence F is chordal by A3, Th48;
  end;
end;
