
theorem
  for G being _finite real-weighted WGraph, W1,W2 being DPath of G, x,y
being set st W1 is_mincost_DPath_from x,y & W2 is_mincost_DPath_from x,y holds
  W1.cost() = W2.cost()
proof
  let G be _finite real-weighted WGraph, W1,W2 be DPath of G, x,y be set;
  assume that
A1: W1 is_mincost_DPath_from x,y and
A2: W2 is_mincost_DPath_from x,y;
  W2 is_Walk_from x,y by A2;
  then
A3: W1.cost() <= W2.cost() by A1;
  W1 is_Walk_from x,y by A1;
  then W2.cost() <= W1.cost() by A2;
  hence thesis by A3,XXREAL_0:1;
end;
