
theorem
  for G1, G2 being _Graph, G3 being SimpleGraph of G1
  st G1 == G2 holds G3 is SimpleGraph of G2
proof
  let G1, G2 be _Graph, G3 be SimpleGraph of G1;
  consider E being RepEdgeSelection of G1 such that
    A1: G3 is inducedSubgraph of G1,the_Vertices_of G1,E\G1.loops() by Def9;
  assume A2: G1 == G2;
  then A3: the_Vertices_of G1 = the_Vertices_of G2 &
    the_Edges_of G1 = the_Edges_of G2 by GLIB_000:def 34;
  then A4: G3 is inducedSubgraph of G2, the_Vertices_of G2, E\G1.loops()
    by A1, A2, GLIB_000:95;
  A5: G1.loops() = G2.loops() by A2, Th50;
  G2 is Subgraph of G1 by A2, GLIB_000:87;
  then E is RepEdgeSelection of G2 by A3, Th78;
  hence thesis by A4, A5, Def9;
end;
