
theorem Th11:
  for G being Dcomplete _Graph, v being Vertex of G holds
    the_Vertices_of G \ {v} c= v.inNeighbors() &
    the_Vertices_of G \ {v} c= v.outNeighbors() &
    the_Vertices_of G \ {v} c= v.allNeighbors()
proof
  let G being Dcomplete _Graph, v be Vertex of G;
  A1: now
    let x be object;
    assume x in the_Vertices_of G \ {v};
    then A2: x in the_Vertices_of G & x <> v by ZFMISC_1:56;
    then reconsider w = x as Vertex of G;
    ex e being object st e DJoins w,v,G by A2, Def1;
    hence x in v.inNeighbors() by GLIB_000:69;
    consider e being object such that
      A3: e DJoins v,w,G by A2, Def1;
    thus x in v.outNeighbors() by A3, GLIB_000:70;
    e Joins v,w,G by A3, GLIB_000:16;
    hence x in v.allNeighbors() by GLIB_000:71;
  end;
  then for x being object st x in the_Vertices_of G \ {v}
    holds x in v.inNeighbors();
  hence the_Vertices_of G \ {v} c= v.inNeighbors() by TARSKI:def 3;
  for x being object st x in the_Vertices_of G \ {v}
    holds x in v.outNeighbors() by A1;
  hence the_Vertices_of G \ {v} c= v.outNeighbors() by TARSKI:def 3;
  for x being object st x in the_Vertices_of G \ {v}
    holds x in v.allNeighbors() by A1;
  hence the_Vertices_of G \ {v} c= v.allNeighbors() by TARSKI:def 3;
end;
