
theorem Th26:
  for G being _finite real-weighted WGraph, s being Vertex of G
  holds dom (DIJK:SSSP(G,s))`1 = G.reachableDFrom(s)
proof
  let G be _finite real-weighted WGraph, src be Vertex of G;
  set Gn = DIJK:SSSP(G, src), RFS = G.reachableDFrom(src);
  set DCS = DIJK:CompSeq(src), n = DCS.Lifespan();
A1: card (dom Gn`1) = min(n+1, card RFS) by Th21
    .= min(card RFS, card RFS) by Th25
    .= card RFS;
  now
    assume
A2: dom Gn`1 <> RFS;
    dom Gn`1 c= RFS by Th19;
    then dom Gn`1 c< RFS by A2,XBOOLE_0:def 8;
    hence contradiction by A1,TREES_1:6;
  end;
  hence thesis;
end;
