let X be set ; :: thesis: ( X = _E implies ( X is Function-like & X is Relation-like ) )

assume A3: X = _E ; :: thesis: ( X is Function-like & X is Relation-like )

consider f, g being Function such that

A4: F = [f,g] and

( dom f c= the_Vertices_of G1 & rng f c= the_Vertices_of G2 & dom g c= the_Edges_of G1 & rng g c= the_Edges_of G2 & ( for e being object st e in dom g holds

( (the_Source_of G1) . e in dom f & (the_Target_of G1) . e in dom f ) ) & ( for e, v, w being object st e in dom g & v in dom f & w in dom f & e Joins v,w,G1 holds

g . e Joins f . v,f . w,G2 ) ) by Def8;

thus ( X is Function-like & X is Relation-like ) by A3, A4; :: thesis: verum

assume A3: X = _E ; :: thesis: ( X is Function-like & X is Relation-like )

consider f, g being Function such that

A4: F = [f,g] and

( dom f c= the_Vertices_of G1 & rng f c= the_Vertices_of G2 & dom g c= the_Edges_of G1 & rng g c= the_Edges_of G2 & ( for e being object st e in dom g holds

( (the_Source_of G1) . e in dom f & (the_Target_of G1) . e in dom f ) ) & ( for e, v, w being object st e in dom g & v in dom f & w in dom f & e Joins v,w,G1 holds

g . e Joins f . v,f . w,G2 ) ) by Def8;

thus ( X is Function-like & X is Relation-like ) by A3, A4; :: thesis: verum