
theorem
  for G1 being simple _Graph, G2 being _Graph
  holds G1 == G2 iff G2 is SimpleGraph of G1
proof
  let G1 be simple _Graph, G2 be _Graph;
  hereby
    assume A1: G1 == G2;
    set E = the RepEdgeSelection of G1;
    A2: the_Edges_of G1 = E \ G1.loops() by Th74;
    G1 is inducedSubgraph of G1, the_Vertices_of G1 by GLIB_000:100;
    then G1 is inducedSubgraph of G1, the_Vertices_of G1, the_Edges_of G1
      by GLIB_000:34;
    then G2 is inducedSubgraph of G1, the_Vertices_of G1, the_Edges_of G1
      by A1, GLIB_000:101;
    hence G2 is SimpleGraph of G1 by A2, Def9;
  end;
  assume G2 is SimpleGraph of G1;
  then consider E being RepEdgeSelection of G1 such that
    A3: G2 is inducedSubgraph of G1,the_Vertices_of G1,E\G1.loops() by Def9;
  the_Edges_of G1 = E \ G1.loops() by Th74;
  then G2 is inducedSubgraph of G1, the_Vertices_of G1 by A3, GLIB_000:34;
  hence G1 == G2 by GLIB_000:94;
end;
