
theorem
  for G1 being non _trivial connected _Graph
  for G2 being non spanning Subgraph of G1
  ex v,e,w being object
  st v <> w & e DJoins v,w,G1 & not e in the_Edges_of G2 &
    (for G3 being addAdjVertex of G2,v,e,w holds G3 is Subgraph of G1) &
    ((v in the_Vertices_of G2 & not w in the_Vertices_of G2) or
      (not v in the_Vertices_of G2 & w in the_Vertices_of G2))
proof
  let G1 be non _trivial connected _Graph, G2 be non spanning Subgraph of G1;
  set S = the_Vertices_of G2;
  A1: S is non empty proper Subset of the_Vertices_of G1
    by GLIB_000:def 33, SUBSET_1:def 6;
  set v0 = the Element of G1.AdjacentSet(S);
  reconsider v0 as Vertex of G1 by A1, TARSKI:def 3;
  consider w0 being Vertex of G1 such that
    A2: w0 in S & v0,w0 are_adjacent by A1, CHORD:50;
  consider e being object such that
    A3: e Joins v0,w0,G1 by A2, CHORD:def 3;
  set v = (the_Source_of G1).e, w = (the_Target_of G1).e;
  take v,e,w;
  A4: e in the_Edges_of G1 by A3, GLIB_000:def 13;
  then A5: e DJoins v,w,G1 & e Joins v,w,G1 by GLIB_000:def 13, def 14;
  then A6: v = v0 & w = w0 or v = w0 & w = v0 by A3, GLIB_000:15;
  A7: not v0 in S by A1, CHORD:50;
  hence v <> w & e DJoins v,w,G1 by A2, A5, A6;
  thus A8: not e in the_Edges_of G2
  proof
    assume e in the_Edges_of G2;
    then e Joins v,w,G2 by A5, GLIB_000:73;
    hence contradiction by A6, A7, GLIB_000:13;
  end;
  hereby
    let G3 be addAdjVertex of G2,v,e,w;
    now
      v in the_Vertices_of G1 & w in the_Vertices_of G1 by A5, GLIB_000:13;
      then A9: {v,w} c= the_Vertices_of G1 by ZFMISC_1:32;
      A10: the_Vertices_of G2 \/ {v,w} c= the_Vertices_of G1 by A9, XBOOLE_1:8;
      the_Vertices_of G3 c= the_Vertices_of G2 \/ {v,w} by Th54;
      hence the_Vertices_of G3 c= the_Vertices_of G1 by A10, XBOOLE_1:1;
      A11: {e} c= the_Edges_of G1 by A4, ZFMISC_1:31;
      A12: the_Edges_of G2 \/ {e} c= the_Edges_of G1 by A11, XBOOLE_1:8;
      A13: the_Edges_of G3 c= the_Edges_of G2 \/ {e} by Th54;
      hence the_Edges_of G3 c= the_Edges_of G1 by A12, XBOOLE_1:1;
      let e0 be set;
      assume e0 in the_Edges_of G3;
      then per cases by A13, ZFMISC_1:136;
      suppose A14: e0 in the_Edges_of G2;
        hence (the_Source_of G3).e0 = (the_Source_of G2).e0 by GLIB_006:def 9
          .= (the_Source_of G1).e0 by A14, GLIB_000:def 32;
        thus (the_Target_of G3).e0
           = (the_Target_of G2).e0 by A14, GLIB_006:def 9
          .= (the_Target_of G1).e0 by A14, GLIB_000:def 32;
      end;
      suppose A15: e0 = e;
        A16: e DJoins v,w,G3 by A2, A6, A7, A8, GLIB_006:131, GLIB_006:132;
        hence (the_Source_of G3).e0 = (the_Source_of G1).e0
          by A15, GLIB_000:def 14;
        thus (the_Target_of G3).e0 = (the_Target_of G1).e0
          by A15, A16, GLIB_000:def 14;
      end;
    end;
    hence G3 is Subgraph of G1 by GLIB_000:def 32;
  end;
  thus thesis by A2, A6, A7;
end;
