
theorem
  for G1 being non _trivial _Graph, v being Vertex of G1
  for G2 being removeVertex of G1, v st v is isolated
  holds G1 is addVertex of G2, v
proof
  let G1 be non _trivial _Graph, v be Vertex of G1;
  let G2 be removeVertex of G1, v;
  assume A1: v is isolated;
  A2: G1 is Supergraph of G2 by GLIB_006:57;
  A3: the_Vertices_of G1 = (the_Vertices_of G1 \ {v}) \/ {v} by ZFMISC_1:116
    .= the_Vertices_of G2 \/ {v} by GLIB_000:47;
  A4: the_Edges_of G1 = G1.edgesBetween(the_Vertices_of G1) by GLIB_000:34
    .= G1.edgesBetween(the_Vertices_of G1 \ {v}) by A1, GLIB_000:108
    .= the_Edges_of G2 by GLIB_000:47;
  A5: dom the_Source_of G1 = the_Edges_of G1 by FUNCT_2:def 1
    .= dom the_Source_of G2 by A4, FUNCT_2:def 1;
  for e being object st e in dom the_Source_of G2 holds
    (the_Source_of G2).e = (the_Source_of G1).e by GLIB_000:def 32;
  then A6: the_Source_of G1 = the_Source_of G2 by A5, FUNCT_1:2;
  A7: dom the_Target_of G1 = the_Edges_of G1 by FUNCT_2:def 1
    .= dom the_Target_of G2 by A4, FUNCT_2:def 1;
  for e being object st e in dom the_Target_of G2 holds
    (the_Target_of G2).e = (the_Target_of G1).e by GLIB_000:def 32;
  then the_Target_of G1 = the_Target_of G2 by A7, FUNCT_1:2;
  hence thesis by A2, A3, A4, A6, GLIB_006:def 10;
end;
