
theorem :: AdjacentV01:
  for G being loopless _Graph, u being Vertex of G holds G.AdjacentSet({
  u}) = {} iff u is isolated
proof
  let G be loopless _Graph, u be Vertex of G;
  hereby
    assume
A1: G.AdjacentSet({u}) = {};
    now
      assume u.edgesInOut() <> {};
      then consider e being object such that
A2:   e in u.edgesInOut() by XBOOLE_0:def 1;
      consider v being Vertex of G such that
A3:   e Joins u,v,G by A2,GLIB_000:64;
A4:   u <> v by A3,GLIB_000:def 18;
      v,u are_adjacent by A3,Def3;
      hence contradiction by A1,A4,Th51;
    end;
    hence u is isolated;
  end;
  assume u is isolated;
  then
A5: u.edgesInOut() = {};
  now
    let v be object such that
A6: v in G.AdjacentSet({u});
    reconsider v as Vertex of G by A6;
    v,u are_adjacent by A6,Th51;
    then ex e being object st e Joins v,u,G;
    hence contradiction by A5,GLIB_000:14,62;
  end;
  hence thesis by XBOOLE_0:def 1;
end;
