let G be _Graph; :: thesis: ( G is loopless iff for v being object holds G .edgesBetween ({v},{v}) = {} )

hereby :: thesis: ( ( for v being object holds G .edgesBetween ({v},{v}) = {} ) implies G is loopless )

assume A1: G is loopless ; :: thesis: for v being object holds G .edgesBetween ({v},{v}) = {}
let v be object ; :: thesis: G .edgesBetween ({v},{v}) = {}
for e being object holds not e in G .edgesBetween ({v},{v})

proof

given e being object such that A2: e in G .edgesBetween ({v},{v}) ; :: thesis: contradiction
e SJoins {v},{v},G by A2, Def30;
hence contradiction by A1, Th137; :: thesis: verum

end;

hence G .edgesBetween ({v},{v}) = {} by XBOOLE_0:def 1; :: thesis: verum

end;

assume A3: for v being object holds G .edgesBetween ({v},{v}) = {} ; :: thesis: G is loopless
for v, e being object holds not e SJoins {v},{v},G

proof

let v be object ; :: thesis: for e being object holds not e SJoins {v},{v},G
given e being object such that A4: e SJoins {v},{v},G ; :: thesis: contradiction
e in G .edgesBetween ({v},{v}) by A4, Def30;
hence contradiction by A3; :: thesis: verum

end;

hence G is loopless by Th137; :: thesis: verum