let G be _Graph; :: thesis: ( G is loopless iff for v, e being object holds not e DJoins v,v,G )

hereby :: thesis: ( ( for v, e being object holds not e DJoins v,v,G ) implies G is loopless )

assume A1: G is loopless ; :: thesis: for v, e being object holds not e DJoins v,v,G
let v be object ; :: thesis: for e being object holds not e DJoins v,v,G
given e being object such that A2: e DJoins v,v,G ; :: thesis: contradiction
e Joins v,v,G by A2, GLIB_000:16;
hence contradiction by A1, GLIB_000:18; :: thesis: verum

end;

assume A3: for v, e being object holds not e DJoins v,v,G ; :: thesis: G is loopless
for v, e being object holds not e Joins v,v,G

proof

let v be object ; :: thesis: for e being object holds not e Joins v,v,G
given e being object such that A4: e Joins v,v,G ; :: thesis: contradiction
e DJoins v,v,G by A4, GLIB_000:16;
hence contradiction by A3; :: thesis: verum

end;

hence G is loopless by GLIB_000:18; :: thesis: verum