reserve x,y,X for set,
  i,j,k,m,n for Nat,
  p for FinSequence of X,
  ii for Integer;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G,
  p,q for oriented Chain of G,
  W for Function,
  U,V,e,ee for set,
  v1,v2,v3,v4 for Vertex of G;
reserve G for finite Graph,
  P,Q for oriented Chain of G,
  v1,v2,v3 for Vertex of G;
reserve G for finite oriented Graph,
  P,Q for oriented Chain of G,
  W for Function of (the carrier' of G), Real>=0,
  v1,v2,v3,v4 for Vertex of G;
reserve f,g,h for Element of REAL*,
  r for Real;
reserve G for oriented Graph,
  v1,v2 for Vertex of G,
  W for Function of (the carrier' of G), Real>=0;

theorem Th25:
  e in the carrier' of G & e orientedly_joins v1,v2 implies Weight
  (v1,v2,W)=W.e
proof
  set EG=the carrier' of G;
  assume
A1: e in EG & e orientedly_joins v1,v2;
  then consider e1 being set such that
A2: XEdge(v1,v2) = e1 and
A3: e1 in EG & e1 orientedly_joins v1,v2 by Def6;
  e=e1 by A1,A3,Th10;
  hence thesis by A2,A3,Def7;
end;
