
theorem
  for G being EGraph, e,x being set holds G == G.labelEdge(e,x)
proof
  let G be EGraph, e,x be set;
  now
    per cases;
    suppose
A1:   e in the_Edges_of G;
A2:   not ELabelSelector in _GraphSelectors by ENUMSET1:def 2;
A3:   G.labelEdge(e,x) = G.set(ELabelSelector, the_ELabel_of G +* (e .-->x
      )) by A1,Def21;
      then
A4:   the_Source_of G = the_Source_of G.labelEdge(e,x) & the_Target_of G =
      the_Target_of G.labelEdge(e,x) by A2,GLIB_000:10;
      the_Vertices_of G = the_Vertices_of G.labelEdge(e,x) & the_Edges_of
      G = the_Edges_of G.labelEdge(e,x) by A3,A2,GLIB_000:10;
      hence thesis by A4;
    end;
    suppose
      not e in the_Edges_of G;
      hence thesis by Def21;
    end;
  end;
  hence thesis;
end;
