reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;
reserve v for object;

theorem Th118:
  for G2 for v1,v2 being Vertex of G2, e being object
  for G1 being addEdge of G2,v1,e,v2
  st G2 is non-Dmulti & not ex e3 being object st e3 DJoins v1,v2,G2
  holds G1 is non-Dmulti
proof
  let G2;
  let v1,v2 be Vertex of G2, e be object;
  let G1 be addEdge of G2,v1,e,v2;
  assume that
    A1: G2 is non-Dmulti and
    A2: not ex e3 being object st e3 DJoins v1,v2,G2;
  per cases;
  suppose A3: v1 in the_Vertices_of G2 & v2 in the_Vertices_of G2 &
    not e in the_Edges_of G2;
    for e1,e2,w1,w2 being object holds e1 DJoins
      w1,w2,G1 & e2 DJoins w1,w2,G1 implies e1 = e2
    proof
      let e1,e2,w1,w2 be object;
      assume that
        A4: e1 DJoins w1,w2,G1 and
        A5: e2 DJoins w1,w2,G1;
      A6: the_Edges_of G1 = the_Edges_of G2 \/ {e} by A3, Def11;
      per cases by A4, Th75;
      suppose A7: e1 DJoins w1,w2,G2;
        per cases by A5, Th75;
        suppose e2 DJoins w1,w2,G2;
          hence thesis by A7, A1, GLIB_000:def 21;
        end;
        suppose A8: not e2 in the_Edges_of G2;
          e2 in the_Edges_of G1 by A5, GLIB_000:def 14;
          then e2 in {e} by A8, A6, XBOOLE_0:def 3;
          then e2 = e by TARSKI:def 1;
          then e2 DJoins v1,v2,G1 by A8, Th109;
          then (the_Source_of G1).e2 = v1 & (the_Target_of G1).e2 = v2
            by GLIB_000:def 14;
          then v1=w1 & v2=w2 by A5, GLIB_000:def 14;
          hence thesis by A2, A7;
        end;
      end;
      suppose A10: not e1 in the_Edges_of G2;
        e1 in the_Edges_of G1 by A4, GLIB_000:def 14;
        then e1 in {e} by A10, A6, XBOOLE_0:def 3;
        then A11: e1 = e by TARSKI:def 1;
        per cases by A5, Th75;
        suppose A12: e2 DJoins w1,w2,G2;
          e1 DJoins v1,v2,G1 by A10, A11, Th109;
          then (the_Source_of G1).e1 = v1 & (the_Target_of G1).e1 = v2
            by GLIB_000:def 14;
          then v1=w1 & v2=w2 by A4, GLIB_000:def 14;
          hence thesis by A2, A12;
        end;
        suppose A13: not e2 in the_Edges_of G2;
          e2 in the_Edges_of G1 by A5, GLIB_000:def 14;
          then e2 in {e} by A13, A6, XBOOLE_0:def 3;
          hence thesis by A11, TARSKI:def 1;
        end;
      end;
    end;
    hence thesis by GLIB_000:def 21;
  end;
  suppose not(v1 in the_Vertices_of G2 & v2 in the_Vertices_of G2 &
    not e in the_Edges_of G2);
    then G1 == G2 by Def11;
    hence thesis by A1, GLIB_000:89;
  end;
end;
