let G be _Graph; :: thesis: for V being non empty one-to-one ManySortedSet of the_Vertices_of G
for E being one-to-one ManySortedSet of the_Edges_of G
for e, v, w being object st e in dom E & v in dom V & w in dom V & E . e Joins V . v,V . w, replaceVerticesEdges (V,E) holds
e Joins v,w,G
let V be non empty one-to-one ManySortedSet of the_Vertices_of G; :: thesis: for E being one-to-one ManySortedSet of the_Edges_of G
for e, v, w being object st e in dom E & v in dom V & w in dom V & E . e Joins V . v,V . w, replaceVerticesEdges (V,E) holds
e Joins v,w,G
let E be one-to-one ManySortedSet of the_Edges_of G; :: thesis: for e, v, w being object st e in dom E & v in dom V & w in dom V & E . e Joins V . v,V . w, replaceVerticesEdges (V,E) holds
e Joins v,w,G
let e, v, w be object ; :: thesis: ( e in dom E & v in dom V & w in dom V & E . e Joins V . v,V . w, replaceVerticesEdges (V,E) implies e Joins v,w,G )
assume A1: ( e in dom E & v in dom V & w in dom V ) ; :: thesis: ( not E . e Joins V . v,V . w, replaceVerticesEdges (V,E) or e Joins v,w,G )
assume E . e Joins V . v,V . w, replaceVerticesEdges (V,E) ; :: thesis: e Joins v,w,G

per cases then ( E . e DJoins V . v,V . w, replaceVerticesEdges (V,E) or E . e DJoins V . w,V . v, replaceVerticesEdges (V,E) ) by GLIB_000:16;

suppose E . e DJoins V . v,V . w, replaceVerticesEdges (V,E) ; :: thesis: e Joins v,w,G

then e DJoins v,w,G by A1, Th10;
hence e Joins v,w,G by GLIB_000:16; :: thesis: verum

end;

suppose E . e DJoins V . w,V . v, replaceVerticesEdges (V,E) ; :: thesis: e Joins v,w,G

then e DJoins w,v,G by A1, Th10;
hence e Joins v,w,G by GLIB_000:16; :: thesis: verum

end;

end;