
theorem
  for F being non empty Graph-yielding Function, x being Element of dom F
  for S being GraphSum of F
  ex M being PGraphMapping of F.x,S st M is strong_SG-embedding
proof
  let F be non empty Graph-yielding Function, x be Element of dom F;
  let S be GraphSum of F;
  set C = canGFDistinction F;
  consider G9 being GraphUnion of rng C such that
    A1: S is G9-Disomorphic by Def27;
  consider M1 being PGraphMapping of G9, S such that
    A2: M1 is Disomorphism by A1, GLIB_010:def 24;
  set H = replaceVerticesEdges(
    renameElementsDistinctlyFunc(the_Vertices_of F,x),
    renameElementsDistinctlyFunc(the_Edges_of F,x));
  set V = the_Vertices_of H;
  x in dom F;
  then H = C.x & x in dom C by Def25;
  then H in rng C by FUNCT_1:3;
  then reconsider H as inducedSubgraph of G9, V by Th62;
  A3: M1 | H is strong_SG-embedding by A2, GLIB_010:58;
  H is F.x-Disomorphic by Th17;
  then consider M2 being PGraphMapping of F.x,H such that
    A4: M2 is Disomorphism by GLIB_010:def 24;
  take (M1 | H) * M2;
  thus thesis by A3, A4, GLIB_010:108;
end;
