
theorem Th123:
  for F being non empty Graph-yielding Function, z being Element of dom F
  ex S being GraphSum of F st S is Supergraph of F.z &
    S is GraphUnion of rng canGFDistinction(F,z)
proof
  let F be non empty Graph-yielding Function, z be Element of dom F;
  set S = the GraphUnion of rng canGFDistinction(F,z);
  set G9 = the GraphUnion of rng canGFDistinction(F);
  set G0 = F.z | _GraphSelectors;
  canGFDistinction(F), canGFDistinction(F,z) are_Disomorphic by Th106;
  then S is G9-Disomorphic by Th55, Th74;
  then reconsider S as GraphSum of F by Def27;
  take S;
  z in dom F;
  then z in dom canGFDistinction(F,z) by Th94;
  then canGFDistinction(F,z).z in rng canGFDistinction(F,z) by FUNCT_1:3;
  then G0 in rng canGFDistinction(F,z) by Th95;
  then A1: G0 is Subgraph of S by GLIB_014:21;
  G0 == F.z by GLIB_000:128;
  then F.z is Subgraph of S by A1, GLIB_000:92;
  hence thesis by GLIB_006:57;
end;
