
theorem Th12:
  for G being loopless Dcomplete _Graph, v being Vertex of G holds
    v.inNeighbors() = the_Vertices_of G \ {v} &
    v.outNeighbors() = the_Vertices_of G \ {v} &
    v.allNeighbors() = the_Vertices_of G \ {v}
proof
  let G be loopless Dcomplete _Graph, v be Vertex of G;
  A1: the_Vertices_of G \ {v} c= v.inNeighbors() &
    the_Vertices_of G \ {v} c= v.outNeighbors() &
    the_Vertices_of G \ {v} c= v.allNeighbors() by Th11;
  A2: v.inNeighbors() c= the_Vertices_of G \ {v} &
    v.outNeighbors() c= the_Vertices_of G \ {v} &
    v.allNeighbors() c= the_Vertices_of G \ {v} by GLIB_000:175;
  thus v.inNeighbors() = the_Vertices_of G \ {v} by A1, A2, XBOOLE_0:def 10;
  thus v.outNeighbors() = the_Vertices_of G \ {v} by A1, A2, XBOOLE_0:def 10;
  thus v.allNeighbors() = the_Vertices_of G \ {v} by A1, A2, XBOOLE_0:def 10;
end;
