let G2 be _Graph; :: thesis: for v1, v2 being Vertex of G2
for e being object
for G1 being addEdge of G2,v1,e,v2 holds the_Vertices_of G1 = the_Vertices_of G2
let v1, v2 be Vertex of G2; :: thesis: for e being object
for G1 being addEdge of G2,v1,e,v2 holds the_Vertices_of G1 = the_Vertices_of G2
let e be object ; :: thesis: for G1 being addEdge of G2,v1,e,v2 holds the_Vertices_of G1 = the_Vertices_of G2
let G1 be addEdge of G2,v1,e,v2; :: thesis: the_Vertices_of G1 = the_Vertices_of G2

per cases ( not e in the_Edges_of G2 or e in the_Edges_of G2 ) ;

suppose not e in the_Edges_of G2 ; :: thesis: the_Vertices_of G1 = the_Vertices_of G2

hence the_Vertices_of G1 = the_Vertices_of G2 by Def11; :: thesis: verum

end;

suppose e in the_Edges_of G2 ; :: thesis: the_Vertices_of G1 = the_Vertices_of G2

then G1 == G2 by Def11;
hence the_Vertices_of G1 = the_Vertices_of G2 by GLIB_000:def 34; :: thesis: verum

end;

end;