
theorem Th45:
  for G being loopless complete _Graph, v being Vertex of G
  holds v.allNeighbors() = the_Vertices_of G \ {v}
proof
  let G be loopless complete _Graph, v be Vertex of G;
  now
    let x be object;
    assume A1: x in v.allNeighbors();
    for e being object holds not e Joins v,v,G by GLIB_000:18;
    then not v in v.allNeighbors() by GLIB_000:71;
    then not x in {v} by A1, TARSKI:def 1;
    hence x in the_Vertices_of G \ {v} by A1, XBOOLE_0:def 5;
  end;
  then A2: v.allNeighbors() c= the_Vertices_of G \ {v} by TARSKI:def 3;
  the_Vertices_of G \ {v} c= v.allNeighbors() by Th44;
  hence thesis by A2, XBOOLE_0:def 10;
end;
