reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem Th1:
  G2 in G1.allSG() iff G2 is plain Subgraph of G1
proof
  hereby
    assume G2 in G1.allSG();
    then consider V being non empty Subset of the_Vertices_of G1,
      E being Subset of the_Edges_of G1 such that
      A1: G2 = the plain inducedSubgraph of G1,V,E and
      E c= G1.edgesBetween(V);
    thus G2 is plain Subgraph of G1 by A1;
  end;
  assume A2: G2 is plain Subgraph of G1;
  set G9 = the plain inducedSubgraph of G1,the_Vertices_of G2,the_Edges_of G2;
  A3: the_Vertices_of G2 c= the_Vertices_of G1 &
    the_Edges_of G2 c= the_Edges_of G1 by A2, GLIB_000:def 32;
  G2.edgesBetween(the_Vertices_of G2) c= G1.edgesBetween(the_Vertices_of G2)
    by A2, GLIB_000:76;
  then A4: the_Edges_of G2 c= G1.edgesBetween(the_Vertices_of G2)
    by GLIB_000:34;
  the_Vertices_of G9 = the_Vertices_of G2 &
    the_Edges_of G9 = the_Edges_of G2 by A3, A4, GLIB_000:def 37;
  then G9 = G2 by A2, GLIB_000:86, GLIB_009:44;
  hence thesis by A3, A4;
end;
