theorem Th23:
  Weight(v1,v2,W) >= 0 iff ex e be set st e in the carrier' of G &
  e orientedly_joins v1,v2
proof
  set EG=the carrier' of G;
  thus Weight(v1,v2,W) >= 0 implies
  ex e be set st e in EG & e orientedly_joins v1,v2 by Def7;
  assume ex e be set st e in the carrier' of G & e orientedly_joins v1,v2;
  then consider e being set such that
A1: XEdge(v1,v2) = e and
A2: e in EG and
A3: e orientedly_joins v1,v2 by Def6;
  e in dom W by A2,FUNCT_2:def 1;
  then W.e in Real>=0 by PARTFUN1:4;
  then ex r being Real st W.e=r & r >=0 by GRAPH_5:def 12;
  hence thesis by A1,A2,A3,Def7;
end;
