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

theorem Th17:
  for G being non edgeless _Graph, e being Edge of G
  holds createGraph(e) is loopless iff not e in G.loops()
proof
  let G be non edgeless _Graph, e be Edge of G;
  hereby
    assume A1: createGraph(e) is loopless;
    assume e in G.loops();
    then consider v being object such that
      A2: e Joins v,v,G by GLIB_009:def 2;
    (the_Source_of G).e = v & (the_Target_of G).e = v by A2, GLIB_000: def 13;
    hence contradiction by A1, Th14, GLIB_000:136;
  end;
  assume A3: not e in G.loops();
  assume not createGraph(e) is loopless;
  then consider v,e0 being object such that
    A4: e0 DJoins v,v,createGraph(e) by GLIB_000:136;
  e0 = e & v is set by A4, Th15, TARSKI:1;
  hence contradiction by A3, A4, GLIB_000:72, GLIB_009:45;
end;
