
theorem
  for G1 being non-multi _Graph, G2 being _Graph
  holds G2 is SimpleGraph of G1 iff G2 is removeLoops of G1
proof
  let G1 be non-multi _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 removeLoops of G1 by A1, Th74;
  end;
  assume A2: G2 is removeLoops of G1;
  set E = the RepEdgeSelection of G1;
  E = the_Edges_of G1 by Th74;
  hence thesis by A2, Def9;
end;
