let G2 be _Graph; :: thesis: for v, e, w being object
for G1 being addEdge of G2,v,e,w holds the_Vertices_of G1 = the_Vertices_of G2
let v, e, w be object ; :: thesis: for G1 being addEdge of G2,v,e,w holds the_Vertices_of G1 = the_Vertices_of G2
let G1 be addEdge of G2,v,e,w; :: thesis: the_Vertices_of G1 = the_Vertices_of G2

per cases ( ( v in the_Vertices_of G2 & w in the_Vertices_of G2 & not e in the_Edges_of G2 ) or not v in the_Vertices_of G2 or not w in the_Vertices_of G2 or e in the_Edges_of G2 ) ;

suppose ( v in the_Vertices_of G2 & w in the_Vertices_of G2 & 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 GLIB_006:def 11; :: thesis: verum

end;

suppose ( not v in the_Vertices_of G2 or not w in the_Vertices_of G2 or e in the_Edges_of G2 ) ; :: thesis: the_Vertices_of G1 = the_Vertices_of G2

then G1 == G2 by GLIB_006:def 11;
hence the_Vertices_of G1 = the_Vertices_of G2 by GLIB_000:def 34; :: thesis: verum

end;

end;