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;

theorem Th9:
  p is_shortestpath_of v1,v2,V,W & q is_shortestpath_of v1,v2,V,W
  implies cost(p,W)=cost(q,W)
proof
  assume that
A1: p is_shortestpath_of v1,v2,V,W and
A2: q is_shortestpath_of v1,v2,V,W;
  q is_orientedpath_of v1,v2,V by A2,GRAPH_5:def 18;
  then
A3: cost(p,W) <= cost(q,W) by A1,GRAPH_5:def 18;
  p is_orientedpath_of v1,v2,V by A1,GRAPH_5:def 18;
  then cost(q,W) <= cost(p,W) by A2,GRAPH_5:def 18;
  hence thesis by A3,XXREAL_0:1;
end;
