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

theorem Th129:
  for G2 for v1 being Vertex of G2, e,v2 being object
  for G1 being addAdjVertex of G2,v1,e,v2
  st not e in the_Edges_of G2 & not v2 in the_Vertices_of G2
  holds ex G3 being addVertex of G2,v2 st G1 is addEdge of G3,v1,e,v2
proof
  let G2;
  let v1 be Vertex of G2;
  let e,v2 be object;
  let G1 be addAdjVertex of G2,v1,e,v2;
  assume A1: not e in the_Edges_of G2 & not v2 in the_Vertices_of G2;
  set G3 = the addVertex of G2,v2;
  take G3;
  A2: the_Vertices_of G1 = the_Vertices_of G2 \/ {v2} &
    the_Edges_of G1 = the_Edges_of G2 \/ {e} &
    the_Source_of G1 = the_Source_of G2 +* (e .--> v1) &
    the_Target_of G1 = the_Target_of G2 +* (e .--> v2) by A1, Def13;
  A3: the_Vertices_of G3 = the_Vertices_of G2 \/ {v2} &
    the_Edges_of G3 = the_Edges_of G2 &
    the_Source_of G3 = the_Source_of G2 &
    the_Target_of G3 = the_Target_of G2 by Def10;
  the_Vertices_of G2 c= the_Vertices_of G3 by Def9;
  then A5: v1 in the_Vertices_of G3 by TARSKI:def 3;
  A6: v2 is Vertex of G3 by Th98;
  A8: the_Edges_of G3 c= the_Edges_of G1 by A2, A3, XBOOLE_1:11;
  for e1 being set st e1 in the_Edges_of G3 holds
    (the_Source_of G3).e1 = (the_Source_of G1).e1 &
    (the_Target_of G3).e1 = (the_Target_of G1).e1 by A3, Def9;
  then G1 is Supergraph of G3 by A2, A3, A8, Def9;
  hence G1 is addEdge of G3,v1,e,v2 by A1, A2, A3, A5, A6, Def11;
end;
