
theorem Th43:
  for G being _finite _Graph, i being Nat, v being set holds ((
  LexBFS:CSeq(G)).i)`2.v c= (Seg G.order()) \ Seg (G.order()-'i)
proof
  let G be _finite _Graph, i be Nat, v be set;
  set CS = (LexBFS:CSeq(G));
  set CSI = CS.i;
  set V2I = CSI`2;
  set CSO = CS.(G.order());
  set V2O = CSO`2;
  defpred P[Nat] means $1 <= G.order() implies (((LexBFS:CSeq(G)).$1)`2).v c=
  (Seg G.order()) \ Seg (G.order() -' $1);
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A2: P[k];
    set CK1 = CS.(k+1);
    set CSK = CS.k;
    set V2K = CSK`2;
    set VLK = CSK`1;
    set V21 = CK1`2;
    per cases;
    suppose
      k+1 <= G.order();
      then
A3:   k < G.order() by NAT_1:13;
      then consider w being Vertex of G such that
      w = LexBFS:PickUnnumbered(CSK) and
A4:   for v being set holds (v in G.AdjacentSet({w}) & not v in dom
VLK implies V21.v = V2K.v \/ {G.order() -' k}) & (not v in G.AdjacentSet({w})
      or v in dom VLK implies V21.v = V2K.v) by Th42;
      per cases;
      suppose
A5:     v in G.AdjacentSet({w}) & not v in dom VLK;
A6:     ((Seg G.order()) \ Seg (G.order() -' k)) \/ {G.order() -' k} = (
        Seg G.order()) \ Seg (G.order() -' (k+1)) by A3,Th5;
        V21.v = V2K.v \/ {G.order() -' k} by A4,A5;
        hence thesis by A2,A6,NAT_1:13,XBOOLE_1:9;
      end;
      suppose
A7:     not v in G.AdjacentSet({w}) or v in dom VLK;
        k <= k+1 by NAT_1:13;
        then
A8:     (Seg G.order()) \ Seg (G.order() -' k) c= (Seg G.order()) \ Seg (
        G.order() -' (k+1)) by Th4;
        V21.v = V2K.v by A4,A7;
        hence thesis by A2,A8,NAT_1:13,XBOOLE_1:1;
      end;
    end;
    suppose
      G.order() < k+1;
      hence thesis;
    end;
  end;
  CS.0 = LexBFS:Init(G) by Def16;
  then (CS.0)`2.v = {};
  then
A9: P[ 0 ] by XBOOLE_1:2;
A10: for k being Nat holds P[k] from NAT_1:sch 2(A9,A1);
  per cases;
  suppose
    i <= G.order();
    hence thesis by A10;
  end;
  suppose
A11: i > G.order();
    then G.order() - i < i - i by XREAL_1:9;
    then G.order() -' i = 0 by XREAL_0:def 2;
    then
A12: G.order() -' G.order() = G.order() -' i by XREAL_1:232;
    V2O = V2I by A11,Th34;
    hence thesis by A10,A12;
  end;
end;
