
theorem Th123:
  for G1 being loopless _Graph, G2 being _Graph
  holds G2 is SimpleGraph of G1 iff G2 is removeParallelEdges of G1
proof
  let G1 be loopless _Graph, G2 be _Graph;
  hereby
    assume G2 is SimpleGraph of G1;
    then consider E being RepEdgeSelection of G1 such that
      A1: G2 is inducedSubgraph of G1, the_Vertices_of G1, E \ G1.loops()
      by Def9;
    thus G2 is removeParallelEdges of G1 by A1, Def7;
  end;
  assume G2 is removeParallelEdges of G1;
  then consider E being RepEdgeSelection of G1 such that
    A2: G2 is inducedSubgraph of G1, the_Vertices_of G1, E by Def7;
  E = E \ G1.loops();
  hence G2 is SimpleGraph of G1 by A2, Def9;
end;
