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

theorem
  for G3 being _Graph, v1, e being object, v2 being Vertex of G3
  for G2 being addVertex of G3,v1, G1 being addEdge of G2,v1,e,v2
  st not e in the_Edges_of G3 & not v1 in the_Vertices_of G3
  holds G1 is addAdjVertex of G3,v1,e,v2
proof
  let G3 be _Graph;
  let v1, e be object;
  let v2 be Vertex of G3;
  let G2 be addVertex of G3,v1;
  let G1 be addEdge of G2,v1,e,v2;
  assume A1: not e in the_Edges_of G3 & not v1 in the_Vertices_of G3;
  A2: G1 is Supergraph of G3 by Th66;
  A3: the_Vertices_of G2 = the_Vertices_of G3 \/ {v1} &
    the_Edges_of G2 = the_Edges_of G3 &
    the_Source_of G2 = the_Source_of G3 &
    the_Target_of G2 = the_Target_of G3 by Def10;
  A5: v2 is Vertex of G2 by Th72;
  v1 is Vertex of G2 by Th98;
  then the_Vertices_of G1 = the_Vertices_of G2 &
    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, A3, A5, Def11;
  hence G1 is addAdjVertex of G3,v1,e,v2 by A1, A2, A3, Def14;
end;
