
theorem
  for G being _Graph
  holds G is non-multi iff EdgeAdjEqRel(G) = id the_Edges_of G
proof
  let G be _Graph;
  hereby
    assume A1: G is non-multi;
    now
      let e1,e2 be object;
      hereby
        assume [e1,e2] in EdgeAdjEqRel(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;
      end;
      assume [e1,e2] in id the_Edges_of G;
      then A4: e1 in the_Edges_of G & e1 = e2 by RELAT_1:def 10;
      now
        reconsider v1 = (the_Source_of G).e1, v2 = (the_Target_of G).e1
          as object;
        take v1,v2;
        thus e1 Joins v1,v2,G & e2 Joins v1,v2,G by A4, GLIB_000:def 13;
      end;
      hence [e1,e2] in EdgeAdjEqRel(G) by Def3;
    end;
    hence EdgeAdjEqRel(G) = id the_Edges_of G by RELAT_1:def 2;
  end;
  assume A5: EdgeAdjEqRel(G) = id the_Edges_of G;
  now
    let e1,e2,v1,v2 be object;
    assume e1 Joins v1,v2,G & e2 Joins v1,v2,G;
    then [e1,e2] in EdgeAdjEqRel(G) by Def3;
    hence e1 = e2 by A5, RELAT_1:def 10;
  end;
  hence thesis by GLIB_000:def 20;
end;
