reserve G for _Graph;

theorem Th26:
  for v, e, w being object, H being addEdge of G,v,e,w
  st ex e0 being object st e0 DJoins v,w,G
  holds VertexDomRel(H) = VertexDomRel(G)
proof
  let v, e, w be object, H be addEdge of G,v,e,w;
  given e0 being object such that
    A1: e0 DJoins v,w,G;
  per cases;
  suppose A2: v in the_Vertices_of G & w in the_Vertices_of G &
      not e in the_Edges_of G;
    G is Subgraph of H by GLIB_006:57;
    then A3: VertexDomRel(G) c= VertexDomRel(H) by Th15;
    now
      let x,y be object;
      assume [x,y] in VertexDomRel(H);
      then consider e9 being object such that
        A4: e9 DJoins x,y,H by Th1;
      per cases by A4, GLIB_006:71;
      suppose e9 DJoins x,y,G;
        hence [x,y] in VertexDomRel(G) by Th1;
      end;
      suppose A5: not e9 in the_Edges_of G;
        A6: the_Edges_of H = the_Edges_of G \/ {e} by A2, GLIB_006:def 11;
        e9 in the_Edges_of H by A4, GLIB_000:def 14;
        then e9 in {e} by A5, A6, XBOOLE_0:def 3;
        then A7: e DJoins x,y,H by A4, TARSKI:def 1;
        e DJoins v,w,H by A2, GLIB_006:105;
        then v = x & w = y by A7, GLIB_000:125;
        hence [x,y] in VertexDomRel(G) by A1, Th1;
      end;
    end;
    then VertexDomRel(H) c= VertexDomRel(G) by RELAT_1:def 3;
    hence thesis by A3, XBOOLE_0:def 10;
  end;
  suppose not(v in the_Vertices_of G & w in the_Vertices_of G &
      not e in the_Edges_of G);
    then G == H by GLIB_006:def 11;
    hence thesis by Th19;
  end;
end;
