reconsider f = (the_Vertices_of H) |` (F _V) as PartFunc of (the_Vertices_of G1),(the_Vertices_of H) by PARTFUN1:12;
reconsider g = (the_Edges_of H) |` (F _E) as PartFunc of (the_Edges_of G1),(the_Edges_of H) by PARTFUN1:12;

now :: thesis: ( ( 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,H ) )

hereby :: thesis: 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,H

let e be object ; :: thesis: ( e in dom g implies ( (the_Source_of G1) . e in dom f & (the_Target_of G1) . e in dom f ) )
assume e in dom g ; :: thesis: ( (the_Source_of G1) . e in dom f & (the_Target_of G1) . e in dom f )
then A1: ( e in dom (F _E) & (F _E) . e in the_Edges_of H ) by FUNCT_1:54;
A2: (F _E) . e Joins (the_Source_of H) . ((F _E) . e),(the_Target_of H) . ((F _E) . e),H by A1, GLIB_000:def 13;
then A3: (F _E) . e Joins (the_Source_of H) . ((F _E) . e),(the_Target_of H) . ((F _E) . e),G2 by GLIB_000:72;
A4: ( (the_Source_of H) . ((F _E) . e) in the_Vertices_of H & (the_Target_of H) . ((F _E) . e) in the_Vertices_of H ) by A2, GLIB_000:13;
A5: e Joins (the_Source_of G1) . e,(the_Target_of G1) . e,G1 by A1, GLIB_000:def 13;
A6: ( (the_Source_of G1) . e in dom (F _V) & (the_Target_of G1) . e in dom (F _V) ) by A1, Th5;
then A7: (F _E) . e Joins (F _V) . ((the_Source_of G1) . e),(F _V) . ((the_Target_of G1) . e),G2 by A1, A5, Th4;
( (F _V) . ((the_Source_of G1) . e) in the_Vertices_of H & (F _V) . ((the_Target_of G1) . e) in the_Vertices_of H )

proof

per cases ( ( (F _V) . ((the_Source_of G1) . e) = (the_Source_of H) . ((F _E) . e) & (F _V) . ((the_Target_of G1) . e) = (the_Target_of H) . ((F _E) . e) ) or ( (F _V) . ((the_Source_of G1) . e) = (the_Target_of H) . ((F _E) . e) & (F _V) . ((the_Target_of G1) . e) = (the_Source_of H) . ((F _E) . e) ) ) by A3, A7, GLIB_000:15;

suppose ( (F _V) . ((the_Source_of G1) . e) = (the_Source_of H) . ((F _E) . e) & (F _V) . ((the_Target_of G1) . e) = (the_Target_of H) . ((F _E) . e) ) ; :: thesis: ( (F _V) . ((the_Source_of G1) . e) in the_Vertices_of H & (F _V) . ((the_Target_of G1) . e) in the_Vertices_of H )

hence ( (F _V) . ((the_Source_of G1) . e) in the_Vertices_of H & (F _V) . ((the_Target_of G1) . e) in the_Vertices_of H ) by A4; :: thesis: verum

end;

suppose ( (F _V) . ((the_Source_of G1) . e) = (the_Target_of H) . ((F _E) . e) & (F _V) . ((the_Target_of G1) . e) = (the_Source_of H) . ((F _E) . e) ) ; :: thesis: ( (F _V) . ((the_Source_of G1) . e) in the_Vertices_of H & (F _V) . ((the_Target_of G1) . e) in the_Vertices_of H )

hence ( (F _V) . ((the_Source_of G1) . e) in the_Vertices_of H & (F _V) . ((the_Target_of G1) . e) in the_Vertices_of H ) by A4; :: thesis: verum

end;

end;

end;

hence ( (the_Source_of G1) . e in dom f & (the_Target_of G1) . e in dom f ) by A6, FUNCT_1:54; :: thesis: verum

end;

let e, v, w be object ; :: thesis: ( e in dom g & v in dom f & w in dom f & e Joins v,w,G1 implies g . e Joins f . v,f . w,H )
assume A8: ( e in dom g & v in dom f & w in dom f ) ; :: thesis: ( e Joins v,w,G1 implies g . e Joins f . v,f . w,H )
then A9: ( e in dom (F _E) & (F _E) . e in the_Edges_of H & v in dom (F _V) & (F _V) . v in the_Vertices_of H & w in dom (F _V) & (F _V) . w in the_Vertices_of H ) by FUNCT_1:54;
then ( e Joins v,w,G1 implies (F _E) . e Joins (F _V) . v,(F _V) . w,G2 ) by Th4;
then A10: ( e Joins v,w,G1 implies (F _E) . e Joins (F _V) . v,(F _V) . w,H ) by A9, GLIB_000:73;
( g . e = (F _E) . e & f . v = (F _V) . v & f . w = (F _V) . w ) by A8, FUNCT_1:53;
hence ( e Joins v,w,G1 implies g . e Joins f . v,f . w,H ) by A10; :: thesis: verum

end;

hence [((the_Vertices_of H) |` (F _V)),((the_Edges_of H) |` (F _E))] is PGraphMapping of G1,H by Th8; :: thesis: verum