let G1, G2 be _Graph; :: thesis: for f being PartFunc of (the_Vertices_of G1),(the_Vertices_of G2)

for g being PartFunc of (the_Edges_of G1),(the_Edges_of G2) st ( 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 DJoins v,w,G1 holds

g . e DJoins f . v,f . w,G2 ) holds

[f,g] is directed PGraphMapping of G1,G2

let f be PartFunc of (the_Vertices_of G1),(the_Vertices_of G2); :: thesis: for g being PartFunc of (the_Edges_of G1),(the_Edges_of G2) st ( 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 DJoins v,w,G1 holds

g . e DJoins f . v,f . w,G2 ) holds

[f,g] is directed PGraphMapping of G1,G2

let g be PartFunc of (the_Edges_of G1),(the_Edges_of G2); :: 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 DJoins v,w,G1 holds

g . e DJoins f . v,f . w,G2 ) implies [f,g] is directed PGraphMapping of G1,G2 )

assume that

A1: 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 ) and

A2: for e, v, w being object st e in dom g & v in dom f & w in dom f & e DJoins v,w,G1 holds

g . e DJoins f . v,f . w,G2 ; :: thesis: [f,g] is directed PGraphMapping of G1,G2

F is directed by A2;

hence [f,g] is directed PGraphMapping of G1,G2 ; :: thesis: verum

for g being PartFunc of (the_Edges_of G1),(the_Edges_of G2) st ( 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 DJoins v,w,G1 holds

g . e DJoins f . v,f . w,G2 ) holds

[f,g] is directed PGraphMapping of G1,G2

let f be PartFunc of (the_Vertices_of G1),(the_Vertices_of G2); :: thesis: for g being PartFunc of (the_Edges_of G1),(the_Edges_of G2) st ( 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 DJoins v,w,G1 holds

g . e DJoins f . v,f . w,G2 ) holds

[f,g] is directed PGraphMapping of G1,G2

let g be PartFunc of (the_Edges_of G1),(the_Edges_of G2); :: 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 DJoins v,w,G1 holds

g . e DJoins f . v,f . w,G2 ) implies [f,g] is directed PGraphMapping of G1,G2 )

assume that

A1: 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 ) and

A2: for e, v, w being object st e in dom g & v in dom f & w in dom f & e DJoins v,w,G1 holds

g . e DJoins f . v,f . w,G2 ; :: thesis: [f,g] is directed PGraphMapping of G1,G2

A3: now :: 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,G2

reconsider F = [f,g] as PGraphMapping of G1,G2 by A1, A3, Th8;g . e Joins f . v,f . w,G2

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 . b_{1} Joins f . b_{2},f . b_{3},G2 )

assume A4: ( e in dom g & v in dom f & w in dom f ) ; :: thesis: ( e Joins v,w,G1 implies g . b_{1} Joins f . b_{2},f . b_{3},G2 )

assume e Joins v,w,G1 ; :: thesis: g . b_{1} Joins f . b_{2},f . b_{3},G2

end;assume A4: ( e in dom g & v in dom f & w in dom f ) ; :: thesis: ( e Joins v,w,G1 implies g . b

assume e Joins v,w,G1 ; :: thesis: g . b

F is directed by A2;

hence [f,g] is directed PGraphMapping of G1,G2 ; :: thesis: verum