reserve G for _Graph;
reserve V for non empty set, E for Relation of V;

theorem Th83:
  for v being Vertex of createGraph(V,E)
  holds v is isolated iff not v in field E
proof
  let v be Vertex of createGraph(V,E);
  hereby
    assume A1: v is isolated;
    assume v in field E;
    then v in dom E \/ rng E by RELAT_1:def 6;
    then per cases by XBOOLE_0:def 3;
    suppose v in dom E;
      then consider w being object such that
        A2: [v,w] in E by XTUPLE_0:def 12;
      thus contradiction by A1, A2, Th63, GLIB_000:144;
    end;
    suppose v in rng E;
      then consider w being object such that
        A3: [w,v] in E by XTUPLE_0:def 13;
      thus contradiction by A1, A3, Th63, GLIB_000:144;
    end;
  end;
  assume A4: not v in field E;
  assume not v is isolated;
  then v.edgesInOut() <> {} by GLIB_000:def 49;
  then consider e being object such that
    A5: e in v.edgesInOut() by XBOOLE_0:def 1;
  e in v.edgesIn() \/ v.edgesOut() by A5, GLIB_000:60;
  then per cases by XBOOLE_0:def 3;
  suppose e in v.edgesIn();
    then consider w being set such that
      A6: e DJoins w,v,createGraph(V,E) by GLIB_000:57;
    e = [w,v] by A6, Th64;
    then [w,v] in E by A6, Th63;
    hence contradiction by A4, RELAT_1:15;
  end;
  suppose e in v.edgesOut();
    then consider w being set such that
      A7: e DJoins v,w,createGraph(V,E) by GLIB_000:59;
    e = [v,w] by A7, Th64;
    then [v,w] in E by A7, Th63;
    hence contradiction by A4, RELAT_1:15;
  end;
end;
