consider f, g being Function such that
A1: [{},{}] = [f,g] and
A2: ( 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 ) and
A3: for e being object st e in dom g holds
( () . e in dom f & () . e in dom f ) and
for e, v, w being object st e in dom g & v in dom f & w in dom f holds
( e Joins v,w,G1 iff g . e Joins f . v,f . w,G2 ) and
A4: 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 Lm2;
take ; :: thesis: ex f, g being Function st
( = [f,g] & 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
( () . e in dom f & () . 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 ) )

thus ex f, g being Function st
( = [f,g] & 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
( () . e in dom f & () . 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 A1, A2, A3, A4; :: thesis: verum