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

theorem Th120:
  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-multi & not v1,v2 are_adjacent
  holds G1 is non-multi
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-multi and
    A2: not v1,v2 are_adjacent;
  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 Joins
      w1,w2,G1 & e2 Joins w1,w2,G1 implies e1 = e2
    proof
      let e1,e2,w1,w2 be object;
      assume that
        A4: e1 Joins w1,w2,G1 and
        A5: e2 Joins w1,w2,G1;
      A6: the_Edges_of G1 = the_Edges_of G2 \/ {e} by A3, Def11;
      per cases by A4, Th76;
      suppose A7: e1 Joins w1,w2,G2;
        per cases by A5, Th76;
        suppose e2 Joins w1,w2,G2;
          hence thesis by A7, A1, GLIB_000:def 20;
        end;
        suppose A8: not e2 in the_Edges_of G2;
          e2 in the_Edges_of G1 by A5, GLIB_000:def 13;
          then e2 in {e} by A8, A6, XBOOLE_0:def 3;
          then e2 = e by TARSKI:def 1;
          then A10: e2 DJoins v1,v2,G1 by A8, Th109;
          per cases by A5, GLIB_000:16;
          suppose e2 DJoins w1,w2,G1;
            then (the_Source_of G1).e2 = w1 & (the_Target_of G1).e2 = w2
              by GLIB_000:def 14;
            then w1=v1 & w2=v2 by A10, GLIB_000:def 14;
            hence thesis by A2, A7, CHORD:def 3;
          end;
          suppose e2 DJoins w2,w1,G1;
            then (the_Source_of G1).e2 = w2 & (the_Target_of G1).e2 = w1
              by GLIB_000:def 14;
            then w2=v1 & w1=v2 by A10, GLIB_000:def 14;
            hence thesis by A2, A7, CHORD:def 3;
          end;
        end;
      end;
      suppose A11: not e1 in the_Edges_of G2;
        e1 in the_Edges_of G1 by A4, GLIB_000:def 13;
        then e1 in {e} by A11, A6, XBOOLE_0:def 3;
        then A12: e1 = e by TARSKI:def 1;
        then A13: e1 DJoins v1,v2,G1 by A11, Th109;
        per cases by A5, Th76;
        suppose A14: e2 Joins w1,w2,G2;
          per cases by A4, GLIB_000:16;
          suppose e1 DJoins w1,w2,G1;
            then (the_Source_of G1).e1 = w1 & (the_Target_of G1).e1 = w2
              by GLIB_000:def 14;
            then w1=v1 & w2=v2 by A13, GLIB_000:def 14;
            hence thesis by A2, A14, CHORD:def 3;
          end;
          suppose e1 DJoins w2,w1,G1;
            then (the_Source_of G1).e1 = w2 & (the_Target_of G1).e1 = w1
              by GLIB_000:def 14;
            then w2=v1 & w1=v2 by A13, GLIB_000:def 14;
            hence thesis by A2, A14, CHORD:def 3;
          end;
        end;
        suppose A15: not e2 in the_Edges_of G2;
          e2 in the_Edges_of G1 by A5, GLIB_000:def 13;
          then e2 in {e} by A15, A6, XBOOLE_0:def 3;
          hence thesis by A12, TARSKI:def 1;
        end;
      end;
    end;
    hence thesis by GLIB_000:def 20;
  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;
