
theorem Th54:
  for G2 being _Graph, v,e,w being object, G1 being addAdjVertex of G2,v,e,w
  holds the_Vertices_of G1 c= the_Vertices_of G2 \/ {v,w} &
    the_Edges_of G1 c= the_Edges_of G2 \/ {e}
proof
  let G2 be _Graph, v,e,w be object, G1 be addAdjVertex of G2,v,e,w;
  per cases;
  suppose v in the_Vertices_of G2 & not w in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    then the_Vertices_of G1 = the_Vertices_of G2 \/ {w} &
      the_Edges_of G1 = the_Edges_of G2 \/ {e} by GLIB_006:def 12;
    hence thesis by ZFMISC_1:7, XBOOLE_1:9;
  end;
  suppose not v in the_Vertices_of G2 & w in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    then the_Vertices_of G1 = the_Vertices_of G2 \/ {v} &
      the_Edges_of G1 = the_Edges_of G2 \/ {e} by GLIB_006:def 12;
    hence thesis by ZFMISC_1:7, XBOOLE_1:9;
  end;
  suppose not((v in the_Vertices_of G2 & not w in the_Vertices_of G2 &
      not e in the_Edges_of G2)or(not v in the_Vertices_of G2 &
      w in the_Vertices_of G2 & not e in the_Edges_of G2));
    then G1 == G2 by GLIB_006:def 12;
    then the_Vertices_of G1 = the_Vertices_of G2 &
      the_Edges_of G1 = the_Edges_of G2 by GLIB_000:def 34;
    hence thesis by XBOOLE_1:7;
  end;
end;
