
theorem Th67:
  for G being _Graph
  holds G.loops() = dom((the_Source_of G) /\ (the_Target_of G))
proof
  let G be _Graph;
  now
    let e be object;
    hereby
      assume e in G.loops();
      then consider v being object such that
        A1: e DJoins v,v,G by GLIB_009:45;
      e in the_Edges_of G by A1, GLIB_000:def 14;
      then A2: e in dom the_Source_of G & e in dom the_Target_of G
        by FUNCT_2:def 1;
      (the_Source_of G).e=v & (the_Target_of G).e=v by A1, GLIB_000:def 14;
      then [e,v] in the_Source_of G & [e,v] in the_Target_of G
        by A2, FUNCT_1:def 2;
      then [e,v] in (the_Source_of G) /\ (the_Target_of G) by XBOOLE_0:def 4;
      hence e in dom((the_Source_of G) /\ (the_Target_of G)) by FUNCT_1:1;
    end;
    set v = ((the_Source_of G) /\ (the_Target_of G)).e;
    assume e in dom((the_Source_of G) /\ (the_Target_of G));
    then [e,v] in (the_Source_of G) /\ (the_Target_of G) by FUNCT_1:def 2;
    then A3: [e,v] in the_Source_of G & [e,v] in the_Target_of G
      by XBOOLE_0:def 4;
    then A4: e in dom the_Source_of G & e in dom the_Target_of G by FUNCT_1:1;
    then (the_Source_of G).e=v & (the_Target_of G).e=v by A3, FUNCT_1:def 2;
    hence e in G.loops() by A4, GLIB_000:def 14, GLIB_009:45;
  end;
  hence thesis by TARSKI:2;
end;
