reserve n,m,i,j,k for Nat,
  x,y,e,X,V,U for set,
  W,f,g for Function;
reserve p,q for FinSequence;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G;
reserve v,v1,v2,v3 for Element of G;
reserve p,q for oriented Chain of G;
reserve G for finite Graph,
  ps for Simple oriented Chain of G,
  P,Q for oriented Chain of G,
  v1,v2,v3 for Element of G,
  pe,qe for FinSequence of the carrier' of G;

theorem
  AcyclicPaths(v1,v2,V) <> {} implies ex pe st pe in AcyclicPaths(v1,v2,
  V) & for qe st qe in AcyclicPaths(v1,v2,V) holds cost(pe,W) <= cost(qe,W) by
