
theorem Th50:
  for G being _finite _Graph holds ((LexBFS:CSeq(G)).Result()`1)"
  is VertexScheme of G
proof
  let G be _finite _Graph;
  set CS = LexBFS:CSeq(G);
  set CSO = CS.(G.order());
  set VLO = CSO`1;
  set VL = CS``1;
A1: CSO = (LexBFS:CSeq(G)).Result() by Th37;
A2: CS.Lifespan() = G.order() by Th37;
A3: VLO = VL.(G.order()) by Def15;
  then
A4: VLO is one-to-one by Th18;
  dom VLO = the_Vertices_of G by Th36;
  then
A5: rng (VLO") = the_Vertices_of G by A4,FUNCT_1:33;
  CS.Lifespan() = VL.Lifespan() by Th39;
  then
  rng (VL.(G.order())) = (Seg G.order()) \ Seg (G.order() -' G.order()) by A2
,Th14
    .= (Seg G.order()) \ Seg 0 by XREAL_1:232
    .= Seg G.order();
  then dom (VLO") = Seg G.order() by A3,A4,FUNCT_1:33;
  then VLO" is FinSequence by FINSEQ_1:def 2;
  then VLO" is FinSequence of the_Vertices_of G by A5,FINSEQ_1:def 4;
  hence thesis by A1,A4,A5,CHORD:def 12;
end;
