let G1, G2, G3, G4 be _Graph; :: thesis: for F being PGraphMapping of G1,G2 st G1 == G3 & G2 == G4 holds

F is PGraphMapping of G3,G4

let F be PGraphMapping of G1,G2; :: thesis: ( G1 == G3 & G2 == G4 implies F is PGraphMapping of G3,G4 )

assume A1: ( G1 == G3 & G2 == G4 ) ; :: thesis: F is PGraphMapping of G3,G4

then A2: ( the_Vertices_of G1 = the_Vertices_of G3 & the_Edges_of G1 = the_Edges_of G3 & the_Vertices_of G2 = the_Vertices_of G4 & the_Edges_of G2 = the_Edges_of G4 ) by GLIB_000:def 34;

then reconsider f = F _V as PartFunc of (the_Vertices_of G3),(the_Vertices_of G4) ;

reconsider g = F _E as PartFunc of (the_Edges_of G3),(the_Edges_of G4) by A2;

hence F is PGraphMapping of G3,G4 ; :: thesis: verum

F is PGraphMapping of G3,G4

let F be PGraphMapping of G1,G2; :: thesis: ( G1 == G3 & G2 == G4 implies F is PGraphMapping of G3,G4 )

assume A1: ( G1 == G3 & G2 == G4 ) ; :: thesis: F is PGraphMapping of G3,G4

then A2: ( the_Vertices_of G1 = the_Vertices_of G3 & the_Edges_of G1 = the_Edges_of G3 & the_Vertices_of G2 = the_Vertices_of G4 & the_Edges_of G2 = the_Edges_of G4 ) by GLIB_000:def 34;

then reconsider f = F _V as PartFunc of (the_Vertices_of G3),(the_Vertices_of G4) ;

reconsider g = F _E as PartFunc of (the_Edges_of G3),(the_Edges_of G4) by A2;

now :: thesis: ( ( for e being object st e in dom g holds

( (the_Source_of G3) . e in dom f & (the_Target_of G3) . 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,G3 holds

g . e Joins f . v,f . w,G4 ) )

assume ( e in dom g & v in dom f & w in dom f ) ; :: thesis: ( e Joins v,w,G3 implies g . e Joins f . v,f . w,G4 )

then ( e Joins v,w,G1 implies g . e Joins f . v,f . w,G2 ) by Th4;

hence ( e Joins v,w,G3 implies g . e Joins f . v,f . w,G4 ) by A1, GLIB_000:88; :: thesis: verum

end;

then
[f,g] is PGraphMapping of G3,G4
by Th8;( (the_Source_of G3) . e in dom f & (the_Target_of G3) . 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,G3 holds

g . e Joins f . v,f . w,G4 ) )

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,G3 holds

g . e Joins f . v,f . w,G4

let e, v, w be object ; :: thesis: ( e in dom g & v in dom f & w in dom f & e Joins v,w,G3 implies g . e Joins f . v,f . w,G4 )g . e Joins f . v,f . w,G4

let e be object ; :: thesis: ( e in dom g implies ( (the_Source_of G3) . e in dom f & (the_Target_of G3) . e in dom f ) )

assume e in dom g ; :: thesis: ( (the_Source_of G3) . e in dom f & (the_Target_of G3) . e in dom f )

then ( (the_Source_of G1) . e in dom f & (the_Target_of G1) . e in dom f ) by Th5;

hence ( (the_Source_of G3) . e in dom f & (the_Target_of G3) . e in dom f ) by A1, GLIB_000:def 34; :: thesis: verum

end;assume e in dom g ; :: thesis: ( (the_Source_of G3) . e in dom f & (the_Target_of G3) . e in dom f )

then ( (the_Source_of G1) . e in dom f & (the_Target_of G1) . e in dom f ) by Th5;

hence ( (the_Source_of G3) . e in dom f & (the_Target_of G3) . e in dom f ) by A1, GLIB_000:def 34; :: thesis: verum

assume ( e in dom g & v in dom f & w in dom f ) ; :: thesis: ( e Joins v,w,G3 implies g . e Joins f . v,f . w,G4 )

then ( e Joins v,w,G1 implies g . e Joins f . v,f . w,G2 ) by Th4;

hence ( e Joins v,w,G3 implies g . e Joins f . v,f . w,G4 ) by A1, GLIB_000:88; :: thesis: verum

hence F is PGraphMapping of G3,G4 ; :: thesis: verum