
theorem
  for G being _finite nonnegative-weighted WGraph, s being Vertex of G,
G2 being inducedWSubgraph of G, dom (DIJK:SSSP(G,s))`1, DIJK:SSSP(G,s)`2 holds
  G2 is_mincost_DTree_rooted_at s & for v being Vertex of G st v in G
  .reachableDFrom(s) holds v in the_Vertices_of G2 & G.min_DPath_cost(s,v) = (
  DIJK:SSSP(G,s))`1.v
proof
  let G be _finite nonnegative-weighted WGraph, src be Vertex of G, G2 be
  inducedWSubgraph of G, dom (DIJK:SSSP(G,src))`1, DIJK:SSSP(G,src)`2;
  set Res = DIJK:SSSP(G,src);
  set dR = dom Res`1;
  thus G2 is_mincost_DTree_rooted_at src by Th23;
  let v being Vertex of G;
  assume v in G.reachableDFrom(src);
  then
A1: v in dR by Th26;
  Res`2 c= G.edgesBetween(dR) by Th22;
  hence v in the_Vertices_of G2 by A1,GLIB_000:def 37;
  thus thesis by A1,Th23;
end;
