
theorem Th65:
  for G being _Graph holds G is edgeless iff
    for v,w being Vertex of G holds not v,w are_adjacent
proof
  let G be _Graph;
  hereby
    assume A1: G is edgeless;
    let v,w be Vertex of G;
    not ex e being object st e Joins v,w,G by A1, GLIB_008:50;
    hence not v,w are_adjacent by CHORD:def 3;
  end;
  assume A2: for v,w being Vertex of G holds not v,w are_adjacent;
  assume G is non edgeless;
  then consider e being object such that
    A3: e in the_Edges_of G by XBOOLE_0:def 1;
  set v = (the_Source_of G).e, w = (the_Target_of G).e;
  reconsider v,w as Vertex of G by A3, FUNCT_2:5;
  e Joins v,w,G by A3, GLIB_000:def 13;
  hence contradiction by A2, CHORD:def 3;
end;
