let G1, G2 be WGraph; :: thesis: for H being WSubgraph of G2

for F being PGraphMapping of G1,G2 st F is weight-preserving holds

H |` F is weight-preserving

let H be WSubgraph of G2; :: thesis: for F being PGraphMapping of G1,G2 st F is weight-preserving holds

H |` F is weight-preserving

let F be PGraphMapping of G1,G2; :: thesis: ( F is weight-preserving implies H |` F is weight-preserving )

assume A1: F is weight-preserving ; :: thesis: H |` F is weight-preserving

(the_Edges_of H) |` (F _E) c= F _E by RELAT_1:86;

then A2: (dom ((the_Edges_of H) |` (F _E))) /\ (dom (F _E)) = dom ((H |` F) _E) by XBOOLE_1:28, RELAT_1:11;

(the_Weight_of H) * ((H |` F) _E) = ((the_Weight_of G2) | (the_Edges_of H)) * ((H |` F) _E) by GLIB_003:def 10

.= (the_Weight_of G2) * ((the_Edges_of H) |` ((the_Edges_of H) |` (F _E))) by GROUP_9:121

.= (the_Weight_of G2) * ((F _E) | (dom ((the_Edges_of H) |` (F _E)))) by GLIB_009:4

.= ((the_Weight_of G1) | (dom (F _E))) | (dom ((H |` F) _E)) by A1, RELAT_1:83

.= (the_Weight_of G1) | (dom ((H |` F) _E)) by A2, RELAT_1:71 ;

hence H |` F is weight-preserving ; :: thesis: verum

for F being PGraphMapping of G1,G2 st F is weight-preserving holds

H |` F is weight-preserving

let H be WSubgraph of G2; :: thesis: for F being PGraphMapping of G1,G2 st F is weight-preserving holds

H |` F is weight-preserving

let F be PGraphMapping of G1,G2; :: thesis: ( F is weight-preserving implies H |` F is weight-preserving )

assume A1: F is weight-preserving ; :: thesis: H |` F is weight-preserving

(the_Edges_of H) |` (F _E) c= F _E by RELAT_1:86;

then A2: (dom ((the_Edges_of H) |` (F _E))) /\ (dom (F _E)) = dom ((H |` F) _E) by XBOOLE_1:28, RELAT_1:11;

(the_Weight_of H) * ((H |` F) _E) = ((the_Weight_of G2) | (the_Edges_of H)) * ((H |` F) _E) by GLIB_003:def 10

.= (the_Weight_of G2) * ((the_Edges_of H) |` ((the_Edges_of H) |` (F _E))) by GROUP_9:121

.= (the_Weight_of G2) * ((F _E) | (dom ((the_Edges_of H) |` (F _E)))) by GLIB_009:4

.= ((the_Weight_of G1) | (dom (F _E))) | (dom ((H |` F) _E)) by A1, RELAT_1:83

.= (the_Weight_of G1) | (dom ((H |` F) _E)) by A2, RELAT_1:71 ;

hence H |` F is weight-preserving ; :: thesis: verum