reserve G,G1,G2 for _Graph;
reserve e,x,y for set;
reserve v,v1,v2 for Vertex of G;
reserve W for Walk of G;

theorem Th1:
  for G being non _trivial connected _Graph, v being Vertex of G
  holds not v is isolated
proof
  let G be non _trivial connected _Graph, v be Vertex of G;
  consider v1,v2 being Vertex of G such that
A1: v1 <> v2 by GLIB_000:21;
  now
    per cases;
    suppose
      v1 = v;
      hence ex u being Vertex of G st u <> v by A1;
    end;
    suppose
      v1 <> v;
      hence ex u being Vertex of G st u <> v;
    end;
  end;
  then consider u being Vertex of G such that
A2: u <> v;
  consider W being Walk of G such that
A3: W is_Walk_from u,v by Def1;
A4: W.first() = u by A3,GLIB_001:def 23;
A5: W.last() = v by A3,GLIB_001:def 23;
  then v in W.vertices() by GLIB_001:88;
  hence thesis by A2,A4,A5,GLIB_001:127,135;
end;
