let G be _Graph; :: thesis: ( G is non-multi iff EdgeAdjEqRel G = id (the_Edges_of G) )

hereby :: thesis: ( EdgeAdjEqRel G = id (the_Edges_of G) implies G is non-multi )

assume A1: G is non-multi ; :: thesis: EdgeAdjEqRel G = id (the_Edges_of G)

now :: thesis: for e1, e2 being object holds
( ( [e1,e2] in EdgeAdjEqRel G implies [e1,e2] in id (the_Edges_of G) ) & ( [e1,e2] in id (the_Edges_of G) implies [e1,e2] in EdgeAdjEqRel G ) )

let e1, e2 be object ; :: thesis: ( ( [e1,e2] in EdgeAdjEqRel G implies [e1,e2] in id (the_Edges_of G) ) & ( [e1,e2] in id (the_Edges_of G) implies [e1,e2] in EdgeAdjEqRel G ) )

hereby :: thesis: ( [e1,e2] in id (the_Edges_of G) implies [e1,e2] in EdgeAdjEqRel G )

assume [e1,e2] in EdgeAdjEqRel G ; :: thesis: [e1,e2] in id (the_Edges_of G)
then consider v1, v2 being object such that
A2: ( e1 Joins v1,v2,G & e2 Joins v1,v2,G ) by Def3;
A3: e1 = e2 by A1, A2, GLIB_000:def 20;
( e1 in the_Edges_of G & e2 in the_Edges_of G ) by A2, GLIB_000:def 13;
hence [e1,e2] in id (the_Edges_of G) by A3, RELAT_1:def 10; :: thesis: verum

end;

assume [e1,e2] in id (the_Edges_of G) ; :: thesis: [e1,e2] in EdgeAdjEqRel G
then A4: ( e1 in the_Edges_of G & e1 = e2 ) by RELAT_1:def 10;

now :: thesis: ex v1, v2 being object st
( e1 Joins v1,v2,G & e2 Joins v1,v2,G )

reconsider v1 = (the_Source_of G) . e1, v2 = (the_Target_of G) . e1 as object ;
take v1 = v1; :: thesis: ex v2 being object st
( e1 Joins v1,v2,G & e2 Joins v1,v2,G )
take v2 = v2; :: thesis: ( e1 Joins v1,v2,G & e2 Joins v1,v2,G )
thus ( e1 Joins v1,v2,G & e2 Joins v1,v2,G ) by A4, GLIB_000:def 13; :: thesis: verum

end;

hence [e1,e2] in EdgeAdjEqRel G by Def3; :: thesis: verum

end;

hence EdgeAdjEqRel G = id (the_Edges_of G) by RELAT_1:def 2; :: thesis: verum

end;

assume A5: EdgeAdjEqRel G = id (the_Edges_of G) ; :: thesis: G is non-multi

now :: thesis: for e1, e2, v1, v2 being object st e1 Joins v1,v2,G & e2 Joins v1,v2,G holds
e1 = e2

let e1, e2, v1, v2 be object ; :: thesis: ( e1 Joins v1,v2,G & e2 Joins v1,v2,G implies e1 = e2 )
assume ( e1 Joins v1,v2,G & e2 Joins v1,v2,G ) ; :: thesis: e1 = e2
then [e1,e2] in EdgeAdjEqRel G by Def3;
hence e1 = e2 by A5, RELAT_1:def 10; :: thesis: verum

end;

hence G is non-multi by GLIB_000:def 20; :: thesis: verum