
theorem Th36:
  for G2, G3 being _Graph, v1,e,v2 being object, G1 being addEdge of G2,v1,e,v2
  st G2 == G3 holds G1 is addEdge of G3,v1,e,v2
proof
  let G2, G3 be _Graph, v1,e,v2 be object;
  let G1 be addEdge of G2,v1,e,v2;
  assume A1: G2 == G3;
  then A2: the_Vertices_of G3 = the_Vertices_of G2 &
    the_Edges_of G3 = the_Edges_of G2 &
    the_Target_of G3 = the_Target_of G2 &
    the_Source_of G3 = the_Source_of G2 by GLIB_000:def 34;
  per cases;
  suppose A3: v1 in the_Vertices_of G2 & v2 in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    then A4: the_Vertices_of G1 = the_Vertices_of G3 &
      the_Edges_of G1 = the_Edges_of G3 \/ {e} &
      the_Source_of G1 = the_Source_of G3 +* (e .--> v1) &
      the_Target_of G1 = the_Target_of G3 +* (e .--> v2)
      by A2, GLIB_006:def 11;
    G2 is Supergraph of G3 by A1, GLIB_006:58;
    then G1 is Supergraph of G3 by GLIB_006:62;
    hence thesis by A2, A3, A4, GLIB_006:def 11;
  end;
  suppose A5: not (v1 in the_Vertices_of G2 & v2 in the_Vertices_of G2 &
      not e in the_Edges_of G2);
    then G1 == G2 by GLIB_006:def 11;
    then A6: G1 == G3 by A1, GLIB_000:85;
    then G1 is Supergraph of G3 by GLIB_006:58;
    hence thesis by A2, A5, A6, GLIB_006:def 11;
  end;
end;
