
theorem Th39:
  for G being _finite connected real-weighted WGraph holds (
  PRIM:MST(G))`1 = the_Vertices_of G
proof
  let G be _finite connected real-weighted WGraph;
  set M = PRIM:MST(G), PCS = PRIM:CompSeq(G), V = M`1;
  set src = the Element of the_Vertices_of G, RFS = G.reachableFrom(src);
  PCS.Lifespan() + 1 = card RFS by Th37;
  then
A1: card V = min(card RFS,card RFS) by Th36;
A2: V c= RFS by Th33;
  now
    assume V <> RFS;
    then V c< RFS by A2,XBOOLE_0:def 8;
    hence contradiction by A1,CARD_2:48;
  end;
  hence thesis by GLIB_002:16;
end;
