let G be _Graph; :: thesis: ( G is loopless iff for v, e being object holds not e SJoins {v},{v},G )
thus ( G is loopless implies for v, e being object holds not e SJoins {v},{v},G ) by Th131, Th18; :: thesis: ( ( for v, e being object holds not e SJoins {v},{v},G ) implies G is loopless )
assume A1: for v, e being object holds not e SJoins {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 A2: e Joins v,v,G ; :: thesis: contradiction
v in {v} by TARSKI:def 1;
then e SJoins {v},{v},G by A2;
hence contradiction by A1; :: thesis: verum

end;

hence G is loopless by Th18; :: thesis: verum