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
  W is_weight>=0of G & P is_shortestpath_of v1,v2,V,W & v1 <> v2 & V c=
U & (for Q, v3 st not v3 in V & Q is_shortestpath_of v1,v3,V,W holds cost(P,W)
  <= cost(Q,W)) implies P is_shortestpath_of v1,v2,U,W
proof
  assume that
A1: W is_weight>=0of G and
A2: P is_shortestpath_of v1,v2,V,W and
A3: v1 <> v2 and
A4: V c= U and
A5: for Q, v3 st not v3 in V & Q is_shortestpath_of v1,v3,V,W holds cost
  (P,W) <= cost(Q,W);
  P is_shortestpath_of v1,v2,W by A1,A2,A3,A5,Th61;
    then
A6: for q be oriented Chain of G st q is_orientedpath_of v1,v2,U holds
    cost(P,W) <= cost(q,W);
  P is_orientedpath_of v1,v2,V by A2;
  then P is_orientedpath_of v1,v2,U by A4,Th30;
  hence thesis by A6;
end;
