
theorem
  for G being VGraph, v,x being set holds G == G.labelVertex(v,x)
proof
  let G be VGraph, e,x be set;
  now
    per cases;
    suppose
A1:   e in the_Vertices_of G;
A2:   not VLabelSelector in _GraphSelectors by ENUMSET1:def 2;
A3:   G.labelVertex(e,x) = G.set(VLabelSelector, the_VLabel_of G +* (e
      .-->x)) by A1,Def22;
      then
A4:   the_Source_of G = the_Source_of G.labelVertex(e,x) & the_Target_of G
      = the_Target_of G.labelVertex(e,x) by A2,GLIB_000:10;
      the_Vertices_of G = the_Vertices_of G.labelVertex(e,x) &
      the_Edges_of G = the_Edges_of G.labelVertex(e,x) by A3,A2,GLIB_000:10;
      hence thesis by A4;
    end;
    suppose
      not e in the_Vertices_of G;
      hence thesis by Def22;
    end;
  end;
  hence thesis;
end;
