let G be _Graph; :: thesis: ( G is non-multi iff for v, w being object holds G .edgesBetween ({v},{w}) is trivial )

hereby :: thesis: ( ( for v, w being object holds G .edgesBetween ({v},{w}) is trivial ) implies G is non-multi )

assume A1: G is non-multi ; :: thesis: for v, w being object holds G .edgesBetween ({v},{w}) is trivial
let v, w be object ; :: thesis: G .edgesBetween ({v},{w}) is trivial
for e1, e2 being object st e1 in G .edgesBetween ({v},{w}) & e2 in G .edgesBetween ({v},{w}) holds
e1 = e2

proof

let e1, e2 be object ; :: thesis: ( e1 in G .edgesBetween ({v},{w}) & e2 in G .edgesBetween ({v},{w}) implies e1 = e2 )
assume e1 in G .edgesBetween ({v},{w}) ; :: thesis: ( not e2 in G .edgesBetween ({v},{w}) or e1 = e2 )
then e1 SJoins {v},{w},G by Def30;
then A2: e1 Joins v,w,G by Th131;
assume e2 in G .edgesBetween ({v},{w}) ; :: thesis: e1 = e2
then e2 SJoins {v},{w},G by Def30;
then e2 Joins v,w,G by Th131;
hence e1 = e2 by A1, A2; :: thesis: verum

end;

hence G .edgesBetween ({v},{w}) is trivial by ZFMISC_1:def 10; :: thesis: verum

end;

assume A3: for v, w being object holds G .edgesBetween ({v},{w}) is trivial ; :: thesis: G is non-multi

now :: thesis: for e1, e2, v, w being object st e1 Joins v,w,G & e2 Joins v,w,G holds
e1 = e2

let e1, e2, v, w be object ; :: thesis: ( e1 Joins v,w,G & e2 Joins v,w,G implies e1 = e2 )
assume A4: ( e1 Joins v,w,G & e2 Joins v,w,G ) ; :: thesis: e1 = e2
( v in {v} & w in {w} ) by TARSKI:def 1;
then ( e1 SJoins {v},{w},G & e2 SJoins {v},{w},G ) by A4;
then A5: ( e1 in G .edgesBetween ({v},{w}) & e2 in G .edgesBetween ({v},{w}) ) by Def30;
G .edgesBetween ({v},{w}) is trivial by A3;
hence e1 = e2 by A5, ZFMISC_1:def 10; :: thesis: verum

end;

hence G is non-multi ; :: thesis: verum