let G be oriented Graph; :: thesis: for v1, v2 being Vertex of G
for W being Function of the carrier' of G,Real>=0 holds
( Weight v1,v2,W >= 0 iff ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) )
let v1, v2 be Vertex of G; :: thesis: for W being Function of the carrier' of G,Real>=0 holds
( Weight v1,v2,W >= 0 iff ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) )
let W be Function of the carrier' of G,Real>=0 ; :: thesis: ( Weight v1,v2,W >= 0 iff ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) )
set EG = the carrier' of G;

hereby :: thesis: ( ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) implies Weight v1,v2,W >= 0 )

assume A1: Weight v1,v2,W >= 0 ; :: thesis: ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 )
assume for e being set holds
( not e in the carrier' of G or not e orientedly_joins v1,v2 ) ; :: thesis: contradiction
then Weight v1,v2,W = - 1 by Def7;
hence contradiction by A1; :: thesis: verum

end;

assume ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) ; :: thesis: Weight v1,v2,W >= 0
then consider e being set such that
A2: ( XEdge v1,v2 = e & e in the carrier' of G & e orientedly_joins v1,v2 ) by Def6;
e in dom W by A2, FUNCT_2:def 1;
then W . e in Real>=0 by PARTFUN1:27;
then consider r being Real such that
A3: ( W . e = r & r >= 0 ) by GRAPH_5:def 12;
thus Weight v1,v2,W >= 0 by A2, A3, Def7; :: thesis: verum